Sequences & Series (Solved MCQs and Notes)

Notes 51 Pages

Joint Entrance Examination (Main)

Mathematics

Contributed by

Nikita Narasimhan

264
Unit - 7
Sequence and Series
Important Points
Sequence
Any function f : N

R is called real sequence.
Any function f : N

C is called complex sequence.
Any function f : {1,2,3, ... n}

X is a finite sequence of a set x (x


).
A sequence is usually written as {f(n)} or {a
n
} or {T
n
} or {t
n
}, f(n) or a
n
or T
n
is called the nth
term of the sequence.
for example 1,
1 1
,
2 3
, ... is a sequence of whose nth term is
1
n
. This sequence is usually
written as
1
 
 
 
n
Series
For any sequence a
1
, a
2
, a
3
.... the sequence {a
1
+a
2
+a
3
+ ... +a
n
} is called a series
i
(a C, i) 
A series is finite or infinite according as the number of terms added is finite or infinite
Progressions (Sequence)
Sequences whose terms follow certain patterns are called progressions
Arithmetic Progression (A.P.)
A sequence a
1
, a
2
, a
3
, ... is said to be an arithmetic progression iff a
n+1
- a
n
= non-zero constant,
for all n. Hear this constant is called the common difference of the A.P. and is usually denoted
by ' d '
A general A.P. is a, a + d, a + 2d, ..., a + (n-1) d... and
n
T = a + (n-1) d
is the general term of
A.P. Hear a is the first term of A.P. and d is the common difference of A.P.
Note that d = T
2
- T
1
= T
3
- T
2
= T
4
- T
3
= ..........
* nth term from the end = l - (n -1) d where l = last term
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265
Sum of the first n terms of an A.P.
S
n
= a + (a+d) + (a + 2d) + ....... + [ a + (n - 1) d ]
=
2
n
[2a + (n - 1) d]
=
2
n
(a+l) where

= T
n
= last term
n = number of terms
a = first term
* Sum of nth term from the end
= [ 2 + (n - 1) d ]l
* If the all terms of an A.P. are increased, decreased, multiplied and divided by the same
non - zero constant, then they remain in A.P.
* In an A.P. sum of terms equidistant from the beginning and end is constant
i.e. a
1
+ a
n
= a
2
+ a
n-1
= a
3
+ a
n-2
= .........
* Three consecutive numbers in A.P. can be taken as a-d, a, a+d
* Four consecutive numbers in A.P. can be taken as a-3d, a-d, a+d, a+3d
* 7Five consecutive numbers in A.P. can be taken as a-2d, a-d, a, a+d, a+2d
* Six consecutive numbers in A.P. can be taken as a-5d, a-3d, a-d, a+d, a+3d,
a+5d.
Arithmetic Means (A.M.)
If a, A, b are in A.P. then A is called by arithmetic mean.
Hear
A =
2
a b
n Arithmetic Mean between a and b
A
1
, A
2
, A
3
, ... A
n
are said to be n A.M.s between two numbers a and b iff a, A
1
, A
2
, ..., A
n
, b are in
A.P. Hence A
1
= a+d, A
2
= a + 2d ... ,A
n
= a + nd
where d =
1


b a
n
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266
* A
1
+ A
2
+ ... + A
n
= n A where A =
2
a b
Harmonic Progression
Non - zero numbers a
1
, a
2
, a
3
, ..., a
n
... are said to be a harmonic progression (H.P.) iff
1 2 3 n
1 1 1 1
, , ,..., ...
a a a a
, are in A.P..
Harmonic mean (H)
If a, H, b are in H.P., then H is called the harmonic mean (H.M.) berween a and b
Hear
2 ab
H =
a+b
n Harmonic mean between a and b
If a, H
1
, H
2
,..., H
n
, b are in H.P., then H
1
, H
2
, ... H
n
are called n harmonic mean between a and b
Geometric Progression (G.P.)
A sequence a
1
, a
2
, a
3
, ..., a
n
, ... of non zero numbers is said to be a geometric progression (G.P.)
iff
32 4 n+1
1 2 3 n
aa a a
= = =...= =
a a a a
a constant for all n

N
This constant is called the common ratio of the G.P. and it is denoted by 'r'
A general G.P. is a, ar, ar
2
, ... ,ar
n-1
,...
nth term of G.P. is
n-1
n
T = ar
Sum of a G.P.
S
n
= sum of first n terms of the G.P.
= a + ar + ar
2
+ ....... + ar
n-1
=
 
1
1
n
r
a
r


if r > 1
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267
=
 
1
1
n
r
a
r


if r < 1
= na if r = 1
S = a + ar + ar
2
+ ... up to infinity
=
1
a
r
where - 1 < r < 1
Geometric Mean (G.M.)
If three positive real numbers a, G, b are in G. P. then G is called the geometric mean between a
and b
Hear
G = ab

G
2
= ab
n Geometric Means
Positive real numbers G
1
, G
2
, G
3
, ..., G
n
are said to be n G.M.s between two positive numbers a
and b iff a, G
1
, G
2
,..., G
n
, b are in G.P.
If r is the common ration of this G.P., then
1
1n
b
r
a
 

 
 
and G
1
= ar, G
2
= ar
2
,..., G
n
= ar
n
Hear G
1
. G
2
. G
3
. .... G
n
=
 
n
2
ab = (ab)
n
=
n
G
* If each term of a G.P. is multiplied or divided by a non - zero number, the resulting progres
sion is also a G.P.
* Three numbers in G.P. can be taken as
a
r
, a, ar
* Four numbers in G.P. can be taken as
3
,
a a
r
r
, ar, ar
3
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268
* Five numbers in G.P. can be taken as
2
,
a a
r
r
, a, ar, ar
2
Arithmetico Geometric Series (A.G.P.)
If P
1
, P
2
, P
3
, ...be an A.P. and a
1
, a
2
, a
3
, ... be a G.P. then p
1
q
1
, p
2
q
2
, p
3
q
3
, ... is said to be an
arithmetico geometric progression. A general A.G.P. is a, (a+d) r, (a+2d)r
2
, (a+3d)r
3
,...
Sum of an A.G.P.
S
n
=
 
 
 
 
 
 
 
n-1
n
n-1
2
dr 1-r
a+(n-1)d r
a
a+ n-1 d r = + - , r 1
1-r 1-r
1-r
 
S

=
im
n

S
n
=
2
1
(1 )
a dr
r
r



, (- 1 < r < 1)
T
n
= n th term of A.G.P. = {a + (n-1)d}r
n-1
Series of natural numbers

n =
1
n
r
r


= 1 + 2 + 3 + ... + n =
( 1)
2
n n 

n
2
=
2
1
n
r
r


= 1
2
+ 2
2
+ 3
2
+ ....+ n
2
=
( 1)(2 1)
6
 n n n

n
3
=
3
1
n
r
r


= 1
3
+ 2
3
+ 3
3
+ ...+ n
3
=
2 2
( 1)
4
n n 
* If the formula of S
n
is given we can obtain the formula for the corresponding sequence
{a
n
} by a
1
= s
1
and

n

2, a
n
= s
n
- s
n-1
* A series is an A.P. iff its nth term is a linear expression in n.
* A sequence is both an A.P. and a G.P. iff it is a constant sequence.
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269
* A series is an A.P. iff S
n
=
2
n
[2a + (n-1)d] is pure quadratic expression in n,
with no constant term.
* A

G

H
* In an A.P. of finitely many terms, sum of terms equidistant from the beginning
and end is constant equal to the sum of the first and last terms.
* In a G.P. of finitely many terms, the product of terms equidistant from the beginning and end
is constant equal to the sum of the first and last terms.
* An A.M. of n real numbers a
1
, a
2
, a
3
, ..., a
n
is A =
1 2 3
...   
n
a a a a
n
* A G.M of n real numbers a
1
, a
2
, a
3
, ..., a
n,
(a
i
> 0, i = 1, 2, ... , n) is
1
n
1 2 n
G (a .a ....a )
* If a
1
, a
2
, a
3
, ... and b
1
, b
2
, b
3
, ... are in G.P.s then a
1
b
1
, a
2
b
2
, a
3
b
3
,...are in G.P. also.
* If a
1
, a
2
, a
3
, ..... a
n
are in G.P. a
i
> 0, i = 1, 2, ...n then
a
2
=

1 3 3 2 4 1 5 4 3 5 2 6
a a , a = a a = a a , a = a a = a a =
1 7 n - 1 n-2 n
a a = ... a = a a
In sort a
r
=
r-k r+k
a a
where k = 0, 1, 2, ... n-r and k

r-1, r = 1 2, ... n-1
Page 6
270
QUESTION BANK
1. If the 1
st
term and common ratio of a G.P. are 1 and 2 respectively then
s
1
+ s
3
+ s
5
+...+ s
2n-1
= __________
(A)
1
3
(2
2n
-5n+4) (B)
1
3
(2
2n+1
-5n)
(C)
1
3
(2
2n+1
-3n-2) (D)
1
3
(2
2n+1
-5n
2
)
2.
1 3 7 15
...100 terms _______
2 4 8 16
    
(A) 2
100
+ 99 (B) 2
-100
+ 99 (C)
101
2 100


(D)
99
2 99


3. If for the triangle whose perimeter is 37 cms and length of sides are in G.P. also the length of
the smallest side is 9 cms then length of remaining two sides are ___ and __
(A) 12, 16 (B) 14, 14 (C) 10, 18 (D) 15, 13
4. Find a, b and c between 2 and 18 such that a+b+c=25, 2,a,b are in A.P. and b,c, 18 are in G.P.
(A) 5, 8, 12 (B) 4, 8, 13 (C) 3, 9, 13 (D) 5, 9, 11
5 Find out four numbers such that, first three numbers are in G.P., last three numbers are in A.P.
having common difference 6, first and last numbers are same.
(A) 8, 4, 2, 8 (B) -8, 4, -2, -8 (C) 8, -4, 2, 8 (D) -8, -4, -2, -8
6. If the A.M. of two numbers a and b is equal to
10
times their G.M. then
a b
a b


=___
(A)
10
3
(B)
3 10
(C)
9
10
(D)
3
10
Page 7
271
7. If the harmonic mean and geometric mean of two numbers a and b are 4 and
3 2
respectively
then the interval [a, b] = _______
(A) [3, 6] (B) [2, 7] (C) [4, 5] (D) [1, 8]
8. A.M of the three numbers which are in G.P. is
14
3
If adding 1 in first and second number and
subtracting 1 from the third number, resulting numbers are in A.P. then the sum of the squares
of original three numbers is ______
(A) 91 (B) 80 (C) 84 (D) 88
9. If the H.M. of a and c is b, G.M. of b and d is c and A.M. of c and e is d, then the G.M. of a and e
is ______
(A) b (B) c (C) d (D) ae
10. If a, b, c are in A.P. and geometric means of ac and ab, ab and bc, ca nad cb are d, e, f respec-
tively then d
2
, e
2
, f
2
are in _____
(A) A. P. (B) G. P. (C) H. P. (D) A. G. P.
11. If two arithmetic means A
1
, A
2
, two geometric means G
1
, G
2
and two harmonic means H
1
, H
2
are inserted between two numbers p and q then _____
(A)
1 2 1 2
1 2 1 2
G G A + A
H H H + H

(B)
1 2 1 2
1 2 1 2
G + G A A
H + H H H

(C)
1 2 1 2
1 2 1 2
G G A - A
H H H - H

(D) (A
1
+ A
2
) (H
1
+ H
2
) = G
1
G
2
H
1
H
2
12.
 
1
2
n
 
= _______
(A) 2n - 2 + 2
1-n
(B) 2n - 2 + 2
n-1
(C) 2n - 2 + 2
-n
(D) 2n - 2 + 2
n+1
13 If (666 ... n times)
2
+ (8888 ... n times) = (4444 ... K times) then K = ______
(A) n + 1 (B) n (C) 2n (D) n
2
Page 8
272
14 If (m +1)
th
, (n +1)
th
and (r +1)
th
terms of an A.P. are in G.P. and m, n, r are in H.P. then the
common difference of the A.P. is _______
(A)
a
n

(B)
2
n
a

(C)
2a
n
(D)
2a
n

15. 2 + 6 + 12 + 20 + 30 + ... 100 terms = ______
(A)
1020300
3
(B)
1030200
3
(C)
1003200
3
(D)
1023200
3
16. If any terms of an A.P. is non - zero and d

0 then
1
r r+1
1
1
a a
n
r



=______
(A)
1 n
n
a a
(B)
1 n
n-1
a a
(C)
1 n
n+1
a a
(D)
1 n
2n
a a
17. If S
1
, S
2
, S
3
, ... , S
n
are the sums of infinite G.P.s. whose first terms ars 1, 2, 3, ..., n and whose
common ratios are
1 1 1 1
, , , ...
2 3 4 n+1
respectively, then
1

n
i
i
S
= ______
(A)
n (n+3)
2
(B)
n (n+4)
2
(C)
n (n-3)
2
(D)
n (n+1)
2
18. 1 + 3 + 7 + 13 + ... 100 terms = _______
(A)
1010000
2
(B)
1000200
3
(C)
1015050
3
(D)
1051050
3
19. 1 + 5 + 14 + 30 + ... n terms = _______
(A)
(n+2) (n+3)
12
(B)
n (n+1) (n+5)
12
(C)
n (n+2) (n+3)
12
(D)
2
n (n+1) (n+2)
12
Page 9
273
20. 4 + 18 + 48 + ... n terms = _______
(A)
n (n+1) (n+2) (3n+5)
12
(B)
n (n+1) (n+2) (5n+3)
12
(C)
n (n+1) (n+2) (7n+1)
12
(D)
n (n+1) (n+2) (9n-1)
12
21. 2 + 12 + 36 + 80 + ... n terms = ______
(A)
n (n+1) (n+2) (3n+5)
24
(B)
n (n+1) (n+2) (3n+1)
12
(C)
n (n+1) (n+3) (n+5)
24
(D)
n (n+1) (n+2) (n+3)
12
22.
3 5 7 9
+ + + ...
4 36 144 400

infinite terms = _______
(A) 0.8 (B) 0.9 (C) 1 (D) 0.99
23.
3 3 3 3 3 3
1 1 +2 1 +2 3
+ +
1 2 3

+ ... up to n terms = ______
(A)
n (n+1) (n+2) (5n+3)
48
(B)
n (n+1) (n+3) (n+5)
24
(C)
n (n+1) (n+2) (7n+1)
48
(D)
n (n+1) (n+2) (3n+5)
48
24.
3 3 3 3 3 3
1 1 +2 1 +2 3
+ +
1 1+2 1+2+3

+ ... 15 terms = _____
(A) 446 (B) 680 (C) 600 (D) 540
Page 10

Sequences & Series (Solved MCQs and Notes)

Sequences & Series (Solved MCQs and Notes)

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