Sequence and series notes

SEQUENCE AND SERIES
CONCEPTS AND RESULTS
** Sequence : is an arrangement of numbers in a definite order according to some rule. A sequence
can also
be defined as a function whose domain is the set of natural numbers or some subsets of the type {1,
2, 3....k). ** A sequence containing a finite number of terms is called a finite sequence. A sequence is
called infinite if it is not a finite sequence.
** Series : If a1, a2, a3,…,an, be a given sequence. Then, the expression a1 + a2 + a3 +,…+ an + ...
** Arithmetic Progression (A.P.) : is a sequence in which terms increase or decrease regularly by the
same
constant.
A sequence a1, a2, a3,…, an,… is called arithmetic sequence or arithmetic progression if
an + 1 = an + d, n ∈ N, where a1 is called the first term and the constant term d is called the common
difference of the A.P.
** The nth term (general term) of the A.P. a, a + d, a + 2d, ... is an = a + (n – 1) d.
** If a, b, c are in A.P. and k(  0) is any constant, then
(i) a + k, b + k, c + k are also in A.P.
(ii) a – k , b – k , c – k are also in A.P.
(iii) ak, bk, ck are also in A.P
a b c
(iv) , , are also in A.P.
k k k
** If a, a + d, a + 2d, …, a + (n – 1) d be an A.P. Then l = a + (n – 1) d.
Sum to n terms Sn  [2a  n  1d
n
2
n
 [a  l]
2
ab
** Arithmetic mean (A.M.) between two numbers a and b is .
2
** n arithmetic means between two numbers a and b are
a
b  a  , a  2b  a  , a  3b  a  , ... , a  n b  a  .
n 1 n 1 n 1 n 1
S
** Sum of n A.M. = n(single A.M.)
** Three consecutive terms in A.P. are a – d , a, a + d.
Four consecutive terms in A.P. are a – 3d , a – d , a + d, a + 3d.
Five consecutive terms in A.P. are a – 2d , a – d , a, a + d, a + 2d.
These results can be used if the sum of the terms is given.
** In an A.P. the sum of terms equidistant from the beginning and end is constant and equal to the sum
of first and last terms.
** mth term from end of an A.P. = (n – m + 1)th term from the beginning.

**Geometric Progression (G . P.) :A sequence is said to be a geometric progression or G.P., if the
ratio of any term to its preceding term is same throughout.
A sequence a1, a2, a3,…, an,… is called geometric progression, if each term is non-zero and
a k 1
 r (constant) , for k ≥ 1.
ak
By taking a1 = a, we obtain a geometric progression, a, ar, ar2, ar3,…., where a is called the first
term
and r is called the common ratio of the G.P.


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, ** General term of a G .P. = a n  ar n 1 .
a (r n  1) a (1  r n )
** Sum to n terms of a G .P. = if r  1 and if r  1 .
r 1 1 r
a
** Sum of terms of an infinite G.P. = .
1 r
** Geometric Mean (G .M.): of two positive numbers a and b is the number is ab .
1 2 3 n
 b  n 1  b  n 1  b  n 1  b  n 1
** n geometric mean between two numbers a and b are a   , a   , a   , ... a   .
a a a a
a
** Three consecutive terms in G.P. are , a , ar .
r
a a
Four consecutive terms in G.P. are 3 , , ar , ar 3 .
r r
a a
Five consecutive terms in G.P. are 2 , , a , ar , ar 2 .
r r
These results can be used if the product of the terms is given.
** Harmonic Progression : A series of quantities is said to be in harmonic progression if their
reciprocals are in arithmetic progression.
1 1 1 1
** n th term of the H.P. , , , ... is .
a a  d a  2d a  n  1d
2ab
** Harmonic Mean between two quantities a & b is
ab
** Relations b/w A(A.M.), G(G.M.) & H(H.M.)
(i) A, G, H are in G.P.
(ii) A, G, H are in descending order of magnitude i.e. A > G > H.
** Arithmetico-geometric series : A type of series in which each term is the product of the
corresponding
terms of an A.P. and a G.P.
a  a  dr  a  2dr 2  a  3dr3  ... is an arithmetico-geometric series.
** Sum to n terms of the arithmetico-geometric series : a  a  dr  a  2dr 2  a  3dr3  ...

is S 
a

 
dr 1  r n 1 [a  n  1d]r n

1 r 1  r 2 1 r
a dr
** Sum of an Infinite arithmetico-geometric series =  .
1  r 1  r 2
** Some useful results
n n  1
 n  1  2  3  ...  n  2
n n  12n  1
 n 2  12  22  32  ...  n 2  6
 n n  1
2

 n  1  2  3  ...  n   2 
3 3 3 3 3




ILLUSTRATIONS

Example 1: A man saves Rs 135/- in the first year, Rs 150/- in the second year and in this way he
increases his savings by Rs 15/- every year. In what time will his total savings be Rs 5550/-?
a) 20 years (b) 25 years (c) 30 years (d) 35 years
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