Important Sequences & Series Formulas for JEE and NEET

 Page 1


Page # 24
SEQUENCE & SERIES
An arithmetic progression (A.P.) : a, a + d, a + 2 d,..... a + (n ? 1) d is an A.P.
Let a be the first term and d be the common difference of an A.P., then n
th
term  = t
n
 = a + (n – 1) d
The sum of first n terms of A.P. are
S
n
 = 
2
n
 [2a + (n – 1) d] = 
2
n
[a + ? ?]
r
th
 term of an A.P. when sum of first r terms is given is t
r
 = S
r
 – S
r
 – 1.
Properties of A.P.
(i) If a, b, c are in A.P. ? ? 2 b = a + c & if a, b, c, d are in A.P.
?  a + d = b + c.
(ii) Three numbers in A.P. can be taken as a ? d, a, a + d; four numbers
in A.P. can be taken as a ? 3d, a ? d, a + d, a + 3d;  five numbers in A.P.
are a ? 2d, a ? d,  a, a + d, a + 2d & six terms in A.P. are a ? 5d,
a ? 3d, a ? d, a + d, a + 3d, a + 5d etc.
(iii) Sum of the terms of an A.P. equidistant from the beginning &
end = sum of first & last term.
Arithmetic Mean (Mean or Average) (A.M.):
If three terms are in A.P. then the middle term is called the A.M. between
the other two, so if a, b, c are in A.P., b is A.M. of a & c.
If a, b are any two given numbers & a, A
1
, A
2
,...., A
n
, b are in A.P. then A
1
, A
2
,...
A
n
 are the
n A.M.’s between a & b. A
1
 = a +
b a
n
?
? 1
,
A
2
 = a +
2
1
(b a)
n
?
?
,......, A A
n
 = a +
 
n b a
n
( ? )
? 1
r
n
?
?
1
A
r
 = nA where A is the single A.M. between a & b.
Page 2


Page # 24
SEQUENCE & SERIES
An arithmetic progression (A.P.) : a, a + d, a + 2 d,..... a + (n ? 1) d is an A.P.
Let a be the first term and d be the common difference of an A.P., then n
th
term  = t
n
 = a + (n – 1) d
The sum of first n terms of A.P. are
S
n
 = 
2
n
 [2a + (n – 1) d] = 
2
n
[a + ? ?]
r
th
 term of an A.P. when sum of first r terms is given is t
r
 = S
r
 – S
r
 – 1.
Properties of A.P.
(i) If a, b, c are in A.P. ? ? 2 b = a + c & if a, b, c, d are in A.P.
?  a + d = b + c.
(ii) Three numbers in A.P. can be taken as a ? d, a, a + d; four numbers
in A.P. can be taken as a ? 3d, a ? d, a + d, a + 3d;  five numbers in A.P.
are a ? 2d, a ? d,  a, a + d, a + 2d & six terms in A.P. are a ? 5d,
a ? 3d, a ? d, a + d, a + 3d, a + 5d etc.
(iii) Sum of the terms of an A.P. equidistant from the beginning &
end = sum of first & last term.
Arithmetic Mean (Mean or Average) (A.M.):
If three terms are in A.P. then the middle term is called the A.M. between
the other two, so if a, b, c are in A.P., b is A.M. of a & c.
If a, b are any two given numbers & a, A
1
, A
2
,...., A
n
, b are in A.P. then A
1
, A
2
,...
A
n
 are the
n A.M.’s between a & b. A
1
 = a +
b a
n
?
? 1
,
A
2
 = a +
2
1
(b a)
n
?
?
,......, A A
n
 = a +
 
n b a
n
( ? )
? 1
r
n
?
?
1
A
r
 = nA where A is the single A.M. between a & b.
Page # 25
Geometric Progression:
a, ar, ar
2
, ar
3
, ar
4
,...... is a G.P. with a as the first term & r as common ratio.
(i) n
th
 term = a
 
r
n ?1
(ii) Sum of the first 
 
n terms i.e. S
n
 =
? ?
?
?
?
?
?
?
?
?
?
1 r , na
1 r ,
1 r
1 r a
n
(iii) Sum of an infinite G.P. when ?r ?
 
< 1 is given by
S
?
 =
? ?
a
r
r
1
1
?
? .
Geometric Means (Mean Proportional) (G.M.):
If a, b, c > 0 are in G.P., b is the G.M. between a & c, then b² = ac
n ?Geometric Means Between positive number a, b: If a, b are two given
numbers & a, G
1
, G
2
,....., G
n
, b are in G.P.. Then G
1
, G
2
, G
3
,...., G
n
 are n
 
G.M.s
between a & b.
G
1
 = a(b/a)
1/n+1
, G
2
 = a(b/a)
2/n+1
,......, G
n
 = a(b/a)
n/n+1
Harmonic Mean (H.M.):
If a, b, c are in H.P., b is the H.M. between a & c, then b = 
c a
ac 2
?
.
H.M. H of a
1
, a
2
 , ........ a
n 
is given by 
H
1
 = 
n
1
 ?
?
?
?
?
?
? ? ?
n 2 1
a
1
.......
a
1
a
1
Page 3


Page # 24
SEQUENCE & SERIES
An arithmetic progression (A.P.) : a, a + d, a + 2 d,..... a + (n ? 1) d is an A.P.
Let a be the first term and d be the common difference of an A.P., then n
th
term  = t
n
 = a + (n – 1) d
The sum of first n terms of A.P. are
S
n
 = 
2
n
 [2a + (n – 1) d] = 
2
n
[a + ? ?]
r
th
 term of an A.P. when sum of first r terms is given is t
r
 = S
r
 – S
r
 – 1.
Properties of A.P.
(i) If a, b, c are in A.P. ? ? 2 b = a + c & if a, b, c, d are in A.P.
?  a + d = b + c.
(ii) Three numbers in A.P. can be taken as a ? d, a, a + d; four numbers
in A.P. can be taken as a ? 3d, a ? d, a + d, a + 3d;  five numbers in A.P.
are a ? 2d, a ? d,  a, a + d, a + 2d & six terms in A.P. are a ? 5d,
a ? 3d, a ? d, a + d, a + 3d, a + 5d etc.
(iii) Sum of the terms of an A.P. equidistant from the beginning &
end = sum of first & last term.
Arithmetic Mean (Mean or Average) (A.M.):
If three terms are in A.P. then the middle term is called the A.M. between
the other two, so if a, b, c are in A.P., b is A.M. of a & c.
If a, b are any two given numbers & a, A
1
, A
2
,...., A
n
, b are in A.P. then A
1
, A
2
,...
A
n
 are the
n A.M.’s between a & b. A
1
 = a +
b a
n
?
? 1
,
A
2
 = a +
2
1
(b a)
n
?
?
,......, A A
n
 = a +
 
n b a
n
( ? )
? 1
r
n
?
?
1
A
r
 = nA where A is the single A.M. between a & b.
Page # 25
Geometric Progression:
a, ar, ar
2
, ar
3
, ar
4
,...... is a G.P. with a as the first term & r as common ratio.
(i) n
th
 term = a
 
r
n ?1
(ii) Sum of the first 
 
n terms i.e. S
n
 =
? ?
?
?
?
?
?
?
?
?
?
1 r , na
1 r ,
1 r
1 r a
n
(iii) Sum of an infinite G.P. when ?r ?
 
< 1 is given by
S
?
 =
? ?
a
r
r
1
1
?
? .
Geometric Means (Mean Proportional) (G.M.):
If a, b, c > 0 are in G.P., b is the G.M. between a & c, then b² = ac
n ?Geometric Means Between positive number a, b: If a, b are two given
numbers & a, G
1
, G
2
,....., G
n
, b are in G.P.. Then G
1
, G
2
, G
3
,...., G
n
 are n
 
G.M.s
between a & b.
G
1
 = a(b/a)
1/n+1
, G
2
 = a(b/a)
2/n+1
,......, G
n
 = a(b/a)
n/n+1
Harmonic Mean (H.M.):
If a, b, c are in H.P., b is the H.M. between a & c, then b = 
c a
ac 2
?
.
H.M. H of a
1
, a
2
 , ........ a
n 
is given by 
H
1
 = 
n
1
 ?
?
?
?
?
?
? ? ?
n 2 1
a
1
.......
a
1
a
1
Page # 26
Relation between 
G² = AH,    A.M. ? G.M. ? H.M.   (only for two numbers)
and   A.M. = G.M. = H.M. if   a
1
 = a
2
 = a
3
 = ...........= a
n
Important Results
(i)
r
n
?
?
1
(a
r
 ± b
r
) = 
r
n
?
?
1
a
r
 ± 
r
n
?
?
1
b
r
.   (ii)
r
n
?
?
1
k a
r
 = k 
r
n
?
?
1
a
r
.
(iii)
r
n
?
?
1
k = nk; where k is a constant.
(iv)
r
n
?
?
1
r = 1 + 2 + 3 +...........+ n =
n n ( ) ?1
2
(v)
r
n
?
?
1
r² = 1
2
 + 2
2
 + 3
2
 +...........+ n
2
 =
n n n ( ) ( ) ? ? 1 2 1
6
(vi)
r
n
?
?
1
r
3
 = 1
3
 + 2
3
 + 3
3
 +...........+ n
3 
=
n n
2 2
1
4
( ) ?