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
( ) ?
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