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Byju's Answer
Standard VIII
Mathematics
Division of an Expression by a Expression
The sum of fi...
Question
The sum of first
n
terms of an infinite G.P. is
A
S
=
a
1
−
r
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B
S
n
=
a
1
(
1
−
r
n
)
1
−
r
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C
S
=
a
n
1
−
r
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D
S
n
=
a
1
(
1
−
r
n
)
1
+
r
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Solution
The correct option is
A
S
=
a
1
−
r
Sum of GP
=
a
(
r
n
−
1
)
r
−
1
a
=
first term
r
=
common ratio
For
n
→
∞
r
n
=
0
for
r
<
1
r
n
→
∞
for
r
>
1
Thus
sum
=
a
r
−
1
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0
Similar questions
Q.
If
S
n
represents the sum of
n
terms of a
G
.
P
.
whose first term and common ratio are
a
and
r
respectively, then prove that
S
1
+
S
2
+
S
3
+
.
.
.
+
S
n
=
n
a
1
−
r
−
a
r
(
1
−
r
n
)
(
1
−
r
)
2
Q.
Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. prove that
P
2
R
n
=
S
n
Q.
For first
n
natural numbers we have the following results with usual notations
n
∑
r
=
1
r
=
n
(
n
+
1
)
2
,
n
∑
r
=
1
r
2
=
n
(
n
+
1
)
(
2
n
+
1
)
6
,
n
∑
r
=
1
r
3
=
(
n
∑
r
=
1
r
)
2
If
a
1
a
2
.
.
.
.
a
n
∈
A
.
P
then sum to
n
terms of the sequence
1
a
1
a
2
,
1
a
2
a
3
,
.
.
.
1
a
n
−
1
a
n
is equal to
n
−
1
a
1
a
n
and the sum to
n
terms of a
G
.
P
with first term '
a
' & common ratio '
r
' is given by
S
n
=
l
r
−
a
r
−
1
for
r
≠
1
for
r
=
1
sum to
n
terms of same
G
.
P
.
is
n
a
, where the sum to infinite terms of
G
.
P
.
is the limiting value of
l
r
−
a
r
−
1
when
n
→
∞
,
|
r
|
<
l
where
l
is the last term of
G
.
P
.
On the basis of above data answer the following questions
The sum to infinite terms of the series
1
2
+
1
6
+
1
18
+
.
.
is equal to ?
Q.
If
S
n
represents the sum of
n
terms of a
G
.
P
.
whose first term and common ratio are
a
and
r
respectively, then prove that
S
1
+
S
3
+
S
5
+
.
.
.
+
S
2
n
−
1
=
a
n
1
−
r
−
a
r
(
1
−
r
2
n
)
(
1
−
r
)
2
(
1
+
r
)
Q.
The set
{
a
1
,
a
2
,
.
.
.
.
a
n
}
form a
G
.
P
.
with first term as
a
and common ratio
r
. Prove that
∑
a
i
a
j
=
a
2
r
(
1
−
r
n
−
1
)
(
1
−
r
n
)
(
1
−
r
)
2
(
1
+
r
)
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