SEQUENCE & SERIES
1. ARITHMETIC PROGRESSION (AP) :
AP is sequence whose terms increase or decrease by a fixed number.
This fixed number is called the common difference. If ‘a’ is the
first term & ‘d’ is the common difference, then AP can be written as
a, a + d, a + 2d, ..., a + (n – 1) d, ...
(a) nth term of this AP Tn = a + (n - 1)d , where d = Tn – Tn–1
n n
(b) The sum of the first n terms : Sn = [2a + (n - 1)d] = [a + l]
2 2
where l is the last term.
(c) Also nth term Tn = Sn - Sn–1
Note :
(i) Sum of first n terms of an A.P. is of the form An2 + Bn i.e. a
quadratic expression in n, in such case the common difference
is twice the coefficient of n2. i.e. 2A
(ii) nth term of an A.P. is of the form An + B i.e. a linear expression
in n, in such case the coefficient of n is the common difference
of the A.P. i.e. A
(iii) Three numbers in AP can be taken as a – d, a, a + d; four
numbers in AP can be taken as a – 3d, a – d, a + d, a + 3d
five numbers in AP are a – 2d, a – d, a, a + d, a + 2d & six
terms in AP are a – 5d, a – 3d, a – d, a + d, a + 3d, a + 5d etc.
a+c
(iv) If a, b, c are in A.P., then b =
2
(v) If a1, a2, a3....... and b1, b2, b3 ......... are two A.P.s, then
a1 ± b1, a2 ± b2, a3 ± b3......... are also in A.P.
(vi) (a) If each term of an A.P. is increased or decreased by the
same number, then the resulting sequence is also an A.P.
having the same common difference.
(b) If each term of an A.P. is multiplied or divided by the same
non zero number (k), then the resulting sequence is also an
A.P. whose common difference is kd & d/k respectively,
where d is common difference of original A.P.
(vii) Any term of an AP (except the first & last) is equal to half the
sum of terms which are equidistant from it.
Tr -k + Tr + k
Tr = , k<r
2
2. GEOMETRIC PROGRESSION (GP) :
GP is a sequence of numbers whose first term is non-zero & each of
the succeeding terms is equal to the preceding terms multiplied by a
constant. Thus in a GP the ratio of successive terms is constant. This
constant factor is called the common ratio of the series & is obtained
by dividing any term by the immediately previous term. Therefore a,
ar, ar2, ar3, ar4, .......... is a GP with 'a' as the first term & 'r' as
common ratio.
(a) nth term Tn = a r n–1
a(r n - 1)
(b) Sum of the first n terms Sn = , if r ¹ l
r -1
(c) Sum of infinite GP when r <1 ( n ® ¥, r n ® 0 )
a
S¥ = ; r <1
1-r
(d) If a, b, c are in GP Þ b2 = ac Þ loga, logb, logc, are in A.P.
Note :
(i) In a G.P. product of kth term from beginning and kth term from the
last is always constant which equal to product of first term and
last term.
(ii) Three numbers in G.P. : a/r, a, ar
Five numbers in G.P. : a/r2, a/r, a, ar, ar2
Four numbers in G.P. : a/r3, a/r, ar, ar3
Six numbers in G.P. : a/r5, a/r3, a/r, ar, ar3, ar5
(iii) If each term of a G.P. be raised to the same power, then resulting
series is also a G.P.
(iv) If each term of a G.P. be multiplied or divided by the same
non-zero quantity, then the resulting sequence is also a G.P.
(v) If a1, a2, a3 ..... and b1, b2, b3, ....... be two G.P.'s of common
ratio r 1 and r 2 respectively, then a 1b 1 , a 2 b 2 .... . and
a1 a2 a3
, , ...... will also form a G.P. common ratio will be r 1 r2
b1 b2 b3
r
and 1 respectively.
r2
(vi) In a positive G.P. every term (except first) is equal to square root
of product of its two terms which are equidistant from it.
i.e. Tr = Tr - k Tr + k , k < r
(vii) If a1, a2, a3.....an is a G.P. of non zero, non negative terms,
then log a1, log a2,.....log an is an A.P. and vice-versa.
3. HARMONIC PROGRESSION (HP) :
A sequence is said to HP if the reciprocals of its terms are in AP.
If the sequence a1, a2, a3, ...., an is an HP then 1/a1, 1/a2,....,
1/an is an AP & vice-versa. Here we do not have the formula for
the sum of the n terms of an HP. The general form of a harmonic
1 1 1 1
progression is , , ,.........
a a + d a + 2d a + ( n - 1) d
Note : No term of any H.P. can be zero. If a, b, c are in
2ac a a-b
HP Þ b = or =
a+c c b-c
4. MEANS
(a) Arithmetic mean (AM) :
If three terms are in AP then the middle term is called the AM
between the other two, so if a, b, c are in AP, b is AM of a & c.
n-arithmetic means between two numbers :
If a,b are any two given numbers & a, A1, A2, ........, An, b are
in AP then A1, A2,...An are the n AM’s between a & b, then
b-a
A1= a + d, A2 = a + 2d ,......, An= a + nd, where d =
n +1
Note : Sum of n AM's inserted between a & b is equal to n times
n
the single AM between a & b i.e. å Ar = nA where A is the
r =1
a+b
single AM between a & b i.e.
2
(b) Geometric mean (GM) :
If a, b, c are in GP, then b is the GM between a & c i.e. b 2 = ac,
therefore b = ac
n-geometric means between two numbers :
If a, b are two given positive numbers & a, G1, G2, ........, Gn,
b are in GP then G1, G2, G3 ,......Gn are n GMs between a & b.
G1= ar, G2 = ar2, ....... Gn= arn, where r= (b/a)1/n+1
Note : The product of n GMs between a & b is equal to nth
n
power of the single GM between a & b i.e. P Gr = (G) n where
r =1
G is the single GM between a & b i.e. ab
(c) Harmonic mean (HM) :
2ac
If a, b, c are in HP, then b is HM between a & c, then b = .
a+c
Important note :
(i) If A, G, H, are respectively AM, GM, HM between two positive
number a & b then
(a) G2 = AH (A, G, H constitute a GP) (b) A ³ G ³ H
(c) A = G = H Û a = b
(ii) Let a1, a2,...... ,an be n positive real numbers, then we define
their arithmetic mean (A), geometric mean (G) and harmonic
a1 + a2 + ..... + a n
mean (H) as A =
n
n
G = (a1 a2.........an)1/n and H =
æ1 1 1 1ö
ç a + a + a + .... + a ÷
è 1 2 3 n ø
It can be shown that A ³ G ³ H. Moreover equality holds at either
place if and only if a1 = a2 =......= an
5. ARITHMETICO - GEOMETRIC SERIES :
Sum of First n terms of an Arithmetico-Geometric Series :
Let Sn = a + (a + d)r + (a + 2d)r 2 + .......... + [a + (n - 1)d]r n -1
a dr(1 - rn-1) [a + (n -1)d] rn
then Sn = + - , r ¹1
1- r (1 - r)2 1- r
Sum to infinity :
a dr
If r < 1 & n ® ¥ then Lim r n = 0 Þ S¥ = +
n®¥ 1 - r (1 - r)2
6. SIGMA NOTATIONS
Theorems :
n n n n n
(a) å (ar ± br ) = å ar ± å br (b) å k ar = k å a r
r =1 r =1 r =1 r =1 r =1
n
(c) å k = nk ; where k is a constant.
r =1
7. RESULTS
n
n(n + 1)
(a) år =
2
(sum of the first n natural numbers)
r =1
n
n(n + 1)(2n + 1)
(b) å r2 =
6
(sum of the squares of the first n natural
r =1
numbers)
2
n
n2 (n + 1)2 én ù
(c) å êå r ú
3
r = = (sum of the cubes of the first n
r =1 4 ër =1 û
natural numbers)
n
n
(d) å r4 = 30 (n + 1)(2n + 1)(3n2 + 3n - 1)
r =1
