HAEF IB - MATH HL
TEST 1
SEQUENCES – BINOMIAL - INDUCTION
by Christos Nikolaidis
Marks:____/100
Name:____________________________________
Date:________________ Grade: ______
Questions
1. [Maximum mark: 6]
Find in terms of a the following results
20 20 20
∑ ak
k =1
∑ (ak + 1)
k =1
∑ ak
k =10
[6 marks]
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1
2. [Maximum mark: 6]
(a) Find the value of sum
39 + 48 + 57 + 66 + L + 246. [3 marks]
n
(b) This sum can be expressed in the form ∑ (ak + b) .
k =1
Find the values of a, b and n. [3 marks]
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2
3. [Maximum mark: 6]
Consider the infinite geometric series
1 1 3
− 2
+ +L
3k 3 6k 12k 3
(a) Find the possible values of k given that the series converges. [4 marks]
(b) Find the exact value of the infinite sum for k = 1. [2 marks]
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3
4. [Maximum mark: 7]
In an arithmetic sequence the sum of the first six terms is 75 and the sum of
the first ten terms is 185.
(a) Find the sum of the first eight terms. [5 marks]
(b) Find the number of terms which are less than 1000. [2 marks]
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4
5. [Maximum mark: 5]
The third term of an infinite geometric sequence is –108 and the fifth term is –72.
Find the possible values for the sum to infinity of the sequence.
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5
6. [Maximum mark: 7]
The value of a share decreases by 2% each day. Its value today is 680€.
Find
(a) the value of the share after 9 days. [2 marks]
(b) the value of the share 10 days ago. [2 marks]
(c) after how many days the value of the share will fall below 400€ [3 marks]
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6
7. [Maximum mark: 8]
(a) In an arithmetic sequence, the sum of the first ten terms is equal to the tenth
term. Find S 9 and u 5 . [5 marks]
(b) The third term of an arithmetic sequence is 7. Find S 5 [3 marks]
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7
8. [Maximum mark: 6]
The sum of the first n terms of a series is given by
S n = n 3 + n where n ∈ +.
(a) Find the first three terms of the sequence and show that it is
neither arithmetic nor geometric. [4 marks]
(b) Find the 10th term of the sequence. [2 marks]
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8
9. [Maximum mark: 8]
(a) Consider the expansion of
8
3 3
x −
x
Find
(i) Find the constant term.
(ii) Find the term in x 4 . [5 marks]
(b) Consider now
8
3 3
x − 3x − 2
2
( )
2
x
Find the term in x 4 . [3 marks]
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9
10. [Maximum mark: 6]
In the expansion of
n
3
x 3 1 −
x
the constant term is – 3240. Find
(a) the value of n. [4 marks]
(b) the coefficient of x. [2 marks]
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10
11. [Maximum mark: 8]
The distinct numbers
x + 9y , x + 3y , x
are the first three terms of a geometric sequence.
(a) Find the common ratio. [5 marks]
(b) Given that the sum to infinity is 48, find the values of x and y. [3 marks]
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11
12. [Maximum mark: 12]
1 2 3 n
Let Sn = + + + L +
2! 3! 4! (n + 1)!
(a) Calculate S1 , S 2 , S 3 , S 4 [2 marks]
(b) Write down the values of 1!, 2!, 3!, 4!, 5! [2 marks]
(c) Hence, guess a formula for S n , for any n ∈ Z + [1 mark]
(d) Show by using mathematical induction that your guess is true. [7 marks]
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12
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13
13. [Maximum mark: 15]
(a) Express ( 3+ 2 ) 3
in the form a 2 + b 3 where a, b ∈ Z + [2 marks]
(b) Express ( 2) ( )
2 4
3+ and 3+ 2 in the form a + b 6 ,
+
where a, b ∈ Z [3 marks]
+
(c) By using mathematical induction, show that, for any n ∈ Z
( 3+ 2 )
2n
has the form a + b 6 , where a, b ∈ Z + [7 marks]
(d) Hence show that, for any n ∈ Z +
( 3+ 2 )
2 n +1
has the form a 2 + b 3 , where a, b ∈ Z + [3 marks]
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14
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15
