S1 Sequences - formula

based
QUESTION PAPER (QP)
1 A geometric progression has first term 64 and sum to infinity 256. Find
(i) the common ratio, [2]
(ii) the sum of the first ten terms. [2]
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2 Find
(i) the sum of the first ten terms of the geometric progression 81, 54, 36, . . . , [3]
(ii) the sum of all the terms in the arithmetic progression 180, 175, 170, . . . , 25. [3]
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3 (a) Find the sum of all the integers between 100 and 400 that are divisible by 7. [4]

(b) The first three terms in a geometric progression are 144, x and 64 respectively, where x is positive.
Find
(i) the value of x,
(ii) the sum to infinity of the progression. [5]
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4 The first term of an arithmetic progression is 6 and the fifth term is 12. The progression has n terms
and the sum of all the terms is 90. Find the value of n. [4]
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5 (a) Find the sum to infinity of the geometric progression with first three terms 0.5, 0.53 and 0.55 .
[3]

(b) The first two terms in an arithmetic progression are 5 and 9. The last term in the progression is
the only term which is greater than 200. Find the sum of all the terms in the progression. [4]

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6 A progression has a second term of 96 and a fourth term of 54. Find the first term of the progression
in each of the following cases:
(i) the progression is arithmetic, [3]
(ii) the progression is geometric with a positive common ratio. [3]

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7 The ninth term of an arithmetic progression is 22 and the sum of the first 4 terms is 49.

(i) Find the first term of the progression and the common difference. [4]

The nth term of the progression is 46.

(ii) Find the value of n. [2]

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8 (a) Find the sum of all the multiples of 5 between 100 and 300 inclusive. [3]

(b) A geometric progression has a common ratio of − 23 and the sum of the first 3 terms is 35. Find
(i) the first term of the progression, [3]
(ii) the sum to infinity. [2]

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9 The first term of a geometric progression is 12 and the second term is −6. Find
(i) the tenth term of the progression, [3]
(ii) the sum to infinity. [2]
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10 (a) The fifth term of an arithmetic progression is 18 and the sum of the first 5 terms is 75. Find the
first term and the common difference. [4]

27
(b) The first term of a geometric progression is 16 and the fourth term is 4. Find the sum to infinity
of the progression. [3]

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11 (a) The first and second terms of an arithmetic progression are 161 and 154 respectively. The sum
of the first m terms is zero. Find the value of m. [3]

(b) A geometric progression, in which all the terms are positive, has common ratio r. The sum of
the first n terms is less than 90% of the sum to infinity. Show that rn > 0.1. [3]
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12 (a) A geometric progression has first term 100 and sum to infinity 2000. Find the second term. [3]

(b) An arithmetic progression has third term 90 and fifth term 80.
(i) Find the first term and the common difference. [2]
(ii) Find the value of m given that the sum of the first m terms is equal to the sum of the first
(m + 1) terms. [2]
(iii) Find the value of n given that the sum of the first n terms is zero. [2]

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13 (a) The first, second and third terms of a geometric progression are 2k + 3, k + 6 and k, respectively.
Given that all the terms of the geometric progression are positive, calculate
(i) the value of the constant k, [3]
(ii) the sum to infinity of the progression. [2]
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14 (a) A geometric progression has a third term of 20 and a sum to infinity which is three times the first
term. Find the first term. [4]

(b) An arithmetic progression is such that the eighth term is three times the third term. Show that
the sum of the first eight terms is four times the sum of the first four terms. [4]
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15 (a) The sixth term of an arithmetic progression is 23 and the sum of the first ten terms is 200. Find
the seventh term. [4]

(b) A geometric progression has first term 1 and common ratio r. A second geometric progression
has first term 4 and common ratio 41 r. The two progressions have the same sum to infinity, S.
Find the values of r and S. [3]
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16 (a) An arithmetic progression contains 25 terms and the first term is −15. The sum of all the terms
in the progression is 525. Calculate
(i) the common difference of the progression, [2]
(ii) the last term in the progression, [2]
(iii) the sum of all the positive terms in the progression. [2]
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17 The first and second terms of a progression are 4 and 8 respectively. Find the sum of the first 10 terms
given that the progression is
(i) an arithmetic progression, [2]
(ii) a geometric progression. [2]
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18 (a) The first two terms of an arithmetic progression are 1 and cos2 x respectively. Show that the sum
of the first ten terms can be expressed in the form a − b sin2 x, where a and b are constants to be
found. [3]

(b) The first two terms of a geometric progression are 1 and 31 tan2 θ respectively, where 0 < θ < 12 π .
(i) Find the set of values of θ for which the progression is convergent. [2]
(ii) Find the exact value of the sum to infinity when θ = 16 π . [2]
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19 The first term of an arithmetic progression is 12 and the sum of the first 9 terms is 135.

(i) Find the common difference of the progression. [2]

The first term, the ninth term and the nth term of this arithmetic progression are the first term, the
second term and the third term respectively of a geometric progression.

(ii) Find the common ratio of the geometric progression and the value of n. [5]

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20 The first term of an arithmetic progression is 61 and the second term is 57. The sum of the first
n terms is n. Find the value of the positive integer n. [4]
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21 (a) In a geometric progression, all the terms are positive, the second term is 24 and the fourth term
is 13 12 . Find
(i) the first term, [3]
(ii) the sum to infinity of the progression. [2]
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22 The first term of a geometric progression is 5 31 and the fourth term is 2 14 . Find
(i) the common ratio, [3]
(ii) the sum to infinity. [2]
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23 The third term of a geometric progression is −108 and the sixth term is 32. Find
(i) the common ratio, [3]
(ii) the first term, [1]
(iii) the sum to infinity. [2]
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24 (a) The first and last terms of an arithmetic progression are 12 and 48 respectively. The sum of the
first four terms is 57. Find the number of terms in the progression. [4]

(b) The third term of a geometric progression is four times the first term. The sum of the first six
terms is k times the first term. Find the possible values of k. [4]

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25 (a) The first 2 terms of a geometric progression are 64 and 48 respectively. The first 3 terms of the
geometric progression are also the 1st term, the 9th term and the nth term respectively of an
arithmetic progression. Find the value of n. [5]
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26 (a) In an arithmetic progression the sum of the first ten terms is 400 and the sum of the next ten
terms is 1000. Find the common difference and the first term. [5]

(b) A geometric progression has first term a, common ratio r and sum to infinity 6. A second
geometric progression has first term 2a, common ratio r2 and sum to infinity 7. Find the values
of a and r. [5]
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27 (a) In a geometric progression, the sum to infinity is equal to eight times the first term. Find the
common ratio. [2]

(b) In an arithmetic progression, the fifth term is 197 and the sum of the first ten terms is 2040. Find
the common difference. [4]
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28 An arithmetic progression has first term a and common difference d. It is given that the sum of the
first 200 terms is 4 times the sum of the first 100 terms.

(i) Find d in terms of a. [3]

(ii) Find the 100th term in terms of a. [2]

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29 The 1st, 2nd and 3rd terms of a geometric progression are the 1st, 9th and 21st terms respectively
of an arithmetic progression. The 1st term of each progression is 8 and the common ratio of the
geometric progression is r, where r ≠ 1. Find
(i) the value of r, [4]
(ii) the 4th term of each progression. [3]
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30 The first term in a progression is 36 and the second term is 32.

(i) Given that the progression is geometric, find the sum to infinity. [2]

(ii) Given instead that the progression is arithmetic, find the number of terms in the progression if
the sum of all the terms is 0. [3]

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31 (i) A geometric progression has first term a (a ≠ 0), common ratio r and sum to infinity S. A second
geometric progression has first term a, common ratio 2r and sum to infinity 3S. Find the value
of r. [3]

(ii) An arithmetic progression has first term 7. The nth term is 84 and the (3n)th term is 245. Find
the value of n. [4]
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32 Three geometric progressions, P, Q and R, are such that their sums to infinity are the first three terms
respectively of an arithmetic progression.

Progression P is 2, 1, 12 , 14 ,  .
Progression Q is 3, 1, 13 , 19 ,  .

(i) Find the sum to infinity of progression R. [3]

(ii) Given that the first term of R is 4, find the sum of the first three terms of R. [3]

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33 (a) The first, second and last terms in an arithmetic progression are 56, 53 and −22 respectively.
Find the sum of all the terms in the progression. [4]

(b) The first, second and third terms of a geometric progression are 2k + 6, 2k and k + 2 respectively,
where k is a positive constant.
(i) Find the value of k. [3]
(ii) Find the sum to infinity of the progression. [2]

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34 (a) The first term of an arithmetic progression is −2222 and the common difference is 17. Find the
value of the first positive term. [3]

(b) The first term of a geometric progression is ï3 and the second term is 2 cos 1, where 0 < 1 < 0.
Find the set of values of 1 for which the progression is convergent. [5]

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35 The first term of a progression is 4x and the second term is x2 .

(i) For the case where the progression is arithmetic with a common difference of 12, find the possible
values of x and the corresponding values of the third term. [4]

(ii) For the case where the progression is geometric with a sum to infinity of 8, find the third term.
[4]
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36 The 12th term of an arithmetic progression is 17 and the sum of the first 31 terms is 1023. Find the
31st term. [5]
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37 (a) The first term of a geometric progression in which all the terms are positive is 50. The third term
is 32. Find the sum to infinity of the progression. [3]

(b) The first three terms of an arithmetic progression are 2 sin x, 3 cos x and sin x + 2 cos x

respectively, where x is an acute angle.

(i) Show that tan x = 43 . [3]

(ii) Find the sum of the first twenty terms of the progression. [3]
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38 (a) A geometric progression is such that the third term is 8 times the sixth term, and the sum of the
first six terms is 31 12 . Find
(i) the first term of the progression, [4]
(ii) the sum to infinity of the progression. [1]
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39 (a) Two convergent geometric progressions, P and Q, have the same sum to infinity. The first and
second terms of P are 6 and 6r respectively. The first and second terms of Q are 12 and −12r
respectively. Find the value of the common sum to infinity. [3]

(b) The first term of an arithmetic progression is cos 1 and the second term is cos 1 + sin2 1, where
0 ≤ 1 ≤ 0. The sum of the first 13 terms is 52. Find the possible values of 1. [5]

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40 (a) An arithmetic progression has a first term of 32, a 5th term of 22 and a last term of −28. Find the
sum of all the terms in the progression. [4]

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41 (a) The first two terms of an arithmetic progression are 16 and 24. Find the least number of terms of
the progression which must be taken for their sum to exceed 20 000. [4]
(b) A geometric progression has a first term of 6 and a sum to infinity of 18. A new geometric
progression is formed by squaring each of the terms of the original progression. Find the sum to
infinity of the new progression. [4]
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42 The common ratio of a geometric progression is r. The first term of the progression is r 2 − 3r + 2

and the sum to infinity is S.

(i) Show that S = 2 − r. [2]

(ii) Find the set of possible values that S can take. [2]

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43 (a) A geometric progression has first term 3a and common ratio r. A second geometric progression
has first term a and common ratio −2r. The two progressions have the same sum to infinity.
Find the value of r. [3]
(b) The first two terms of an arithmetic progression are 15 and 19 respectively. The first two terms
of a second arithmetic progression are 420 and 415 respectively. The two progressions have the
same sum of the first n terms. Find the value of n. [3]
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44 An arithmetic progression has first term −12 and common difference 6. The sum of the first n terms
exceeds 3000. Calculate the least possible value of n. [4]
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45 (a) A geometric progression has a second term of 12 and a sum to infinity of 54. Find the possible
values of the first term of the progression. [4]
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46 The common ratio of a geometric progression is 0.99. Express the sum of the first 100 terms as a
percentage of the sum to infinity, giving your answer correct to 2 significant figures. [5]

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47 The first term of a series is 6 and the second term is 2.

(i) For the case where the series is an arithmetic progression, find the sum of the first 80 terms. [3]

(ii) For the case where the series is a geometric progression, find the sum to infinity. [2]

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48 The first three terms of an arithmetic progression are 4, x and y respectively. The first three terms of
a geometric progression are x, y and 18 respectively. It is given that both x and y are positive.

(i) Find the value of x and the value of y. [4]

(ii) Find the fourth term of each progression. [3]
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49 In an arithmetic progression the first term is a and the common difference is 3. The nth term is 94
and the sum of the first n terms is 1420. Find n and a. [6]
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50 (i) The first and second terms of a geometric progression are p and 2p respectively, where p is a
positive constant. The sum of the first n terms is greater than 1000p. Show that 2n > 1001. [2]

(ii) In another case, p and 2p are the first and second terms respectively of an arithmetic progression.
The nth term is 336 and the sum of the first n terms is 7224. Write down two equations in n and
p and hence find the values of n and p. [5]
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51 The third and fourth terms of a geometric progression are 48 and 32 respectively. Find the sum
to infinity of the progression. [3]
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52 (a) In an arithmetic progression, the sum of the first ten terms is equal to the sum of the next five
terms. The first term is a.
(i) Show that the common difference of the progression is 31 a. [4]

(ii) Given that the tenth term is 36 more than the fourth term, find the value of a. [2]
(b) The sum to infinity of a geometric progression is 9 times the sum of the first four terms. Given
that the first term is 12, find the value of the fifth term. [4]

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53 The first, second and third terms of a geometric progression are x, x − 3 and x − 5 respectively.
(i) Find the value of x. [2]
(ii) Find the fourth term of the progression. [2]
(iii) Find the sum to infinity of the progression. [2]

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54 The first, second and third terms of a geometric progression are 3k, 5k − 6 and 6k − 4, respectively.

(i) Show that k satisfies the equation 7k2 − 48k + 36 = 0. [2]

(ii) Find, showing all necessary working, the exact values of the common ratio corresponding to
each of the possible values of k. [4]
(iii) One of these ratios gives a progression which is convergent. Find the sum to infinity. [2]

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55 The sum of the first nine terms of an arithmetic progression is 117. The sum of the next four terms
is 91.

Find the first term and the common difference of the progression. [4]
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56 The first term of a progression is sin2 1, where 0 < 1 < 12 π. The second term of the progression is
sin2 1 cos2 1.

(a) Given that the progression is geometric, find the sum to infinity. [3]

It is now given instead that the progression is arithmetic.

(b) (i) Find the common difference of the progression in terms of sin 1. [3]

(ii) Find the sum of the first 16 terms when 1 = 13 π. [3]

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57 A geometric progression has first term a, common ratio r and sum to infinity S. A second geometric
progression has first term a, common ratio R and sum to infinity 2S.

(a) Show that r = 2R − 1. [3]

It is now given that the 3rd term of the first progression is equal to the 2nd term of the second
progression.

(b) Express S in terms of a. [4]

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58 The first, second and third terms of a geometric progression are 2p + 6, −2p and p + 2 respectively,
where p is positive.

Find the sum to infinity of the progression. [5]

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59 In the expansion of a + bx
7 , where a and b are non-zero constants, the coefficients of x, x2 and x4
are the first, second and third terms respectively of a geometric progression.

a
Find the value of . [5]
b
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tan2 1
−
1
cos2 1 cos2 1
60 The first and second terms of an arithmetic progression are and , respectively, where
0 < 1 < 12 π.

(a) Show that the common difference is −

1
cos4 1
. [4]

(b) Find the exact value of the 13th term when 1 = 16 π. [3]

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61 The first term of a progression is cos 1, where 0 < 1 < 12 π.

1
(a) For the case where the progression is geometric, the sum to infinity is .
cos 1
(i) Show that the second term is cos 1 sin2 1. [3]

(ii) Find the sum of the first 12 terms when 1 = 13 π, giving your answer correct to 4 significant
figures. [2]

(b) For the case where the progression is arithmetic, the first two terms are again cos 1 and cos 1 sin2 1
respectively.

Find the 85th term when 1 = 13 π. [4]

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62 The sum of the first 20 terms of an arithmetic progression is 405 and the sum of the first 40 terms
is 1410.

Find the 60th term of the progression. [5]

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63 The fifth, sixth and seventh terms of a geometric progression are 8k, −12 and 2k respectively.

Given that k is negative, find the sum to infinity of the progression. [4]

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64 The first, second and third terms of an arithmetic progression are a, 23 a and b respectively, where
a and b are positive constants. The first, second and third terms of a geometric progression are
a, 18 and b + 3 respectively.

(a) Find the values of a and b. [5]

(b) Find the sum of the first 20 terms of the arithmetic progression. [3]
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65 (a) A geometric progression is such that the second term is equal to 24% of the sum to infinity.

Find the possible values of the common ratio. [3]

(b) An arithmetic progression P has first term a and common difference d . An arithmetic progression
Q has first term 2 a + 1
and common difference d + 1
. It is given that
5th term of P 1 Sum of first 5 terms of P 2
= and = .
12th term of Q 3 Sum of first 5 terms of Q 3
Find the value of a and the value of d . [6]

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66 The first term of an arithmetic progression is a and the common difference is −4. The first term
of a geometric progression is 5a and the common ratio is − 14 . The sum to infinity of the geometric
progression is equal to the sum of the first eight terms of the arithmetic progression.

(a) Find the value of a. [4]

The kth term of the arithmetic progression is zero.

(b) Find the value of k. [2]

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67 The first, third and fifth terms of an arithmetic progression are 2 cos x, −6 3 sin x and 10 cos x
respectively, where 21 π < x < π.

(a) Find the exact value of x. [3]

(b) Hence find the exact sum of the first 25 terms of the progression. [3]

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68 The second term of a geometric progression is 54 and the sum to infinity of the progression is 243.
The common ratio is greater than 12 .

Find the tenth term, giving your answer in exact form. [5]
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69 The first term of an arithmetic progression is 84 and the common difference is −3.

(a) Find the smallest value of n for which the nth term is negative. [2]

It is given that the sum of the first 2k terms of this progression is equal to the sum of the first k terms.

(b) Find the value of k. [3]

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70 The thirteenth term of an arithmetic progression is 12 and the sum of the first 30 terms is −15.

Find the sum of the first 50 terms of the progression. [5]

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71 The second and third terms of a geometric progression are 10 and 8 respectively.

Find the sum to infinity. [4]

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72 The first, second and third terms of an arithmetic progression are k, 6k and k + 6 respectively.

(a) Find the value of the constant k. [2]

(b) Find the sum of the first 30 terms of the progression. [3]

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73 An arithmetic progression has first term 4 and common difference d . The sum of the first n terms of
the progression is 5863.

11726
(a) Show that n − 1
d = − 8. [1]
n
(b) Given that the nth term is 139, find the values of n and d , giving the value of d as a fraction. [4]

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74 The first, second and third terms of an arithmetic progression are a, 2a and a2 respectively, where a
is a positive constant.

Find the sum of the first 50 terms of the progression. [5]

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75 A geometric progression is such that the third term is 1764 and the sum of the second and third terms
is 3444.

Find the 50th term. [4]

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2
p
76 The first three terms of an arithmetic progression are , 2p − 6 and p.
6

(a) Given that the common difference of the progression is not zero, find the value of p. [3]
(b) Using this value, find the sum to infinity of the geometric progression with first two terms
p2
and 2p − 6. [2]
6
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77 The second term of a geometric progression is 16 and the sum to infinity is 100.

(a) Find the two possible values of the first term. [4]
(b) Show that the nth term of one of the two possible geometric progressions is equal to
4n−2 multiplied by the nth term of the other geometric progression. [4]

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 a2
78 A progression has first term a and second term , where a is a positive constant.
a+2
(a) For the case where the progression is geometric and the sum to infinity is 264, find the value
of a. [5]
(b) For the case where the progression is arithmetic and a = 6, determine the least value of n required
for the sum of the first n terms to be less than −480. [5]

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