If S, denotes the sum of n terms of a G.P., prove that
(S10-S20)2 = S10(S30 - S20).
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Answer:
Since S
n
is the sum of the n terms of a G.P., then,
S
n
=
1−r
a(1−r
n
)
Simplifying the LHS of (S
10
−S
20
)
2
=S
10
(S
30
−S
20
),
(S
10
−S
20
)
2
=(
1−r
a(1−r
10
)
−
1−r
a(1−r
20
)
)
2
=(
1−r
a(1−r
10
)−a(1−r
20
)
)
2
=(
1−r
a−ar
10
−a+ar
20
)
2
=(
1−r
ar
20
−ar
10
)
2
=a
2
(r
10
(
1−r
r
10
−1
))
2
=a
2
r
20
(
1−r
r
10
−1
)
2
=a
2
r
20
(
1−r
1−r
10
)
2
Simplifying the RHS of (S
10
−S
20
)
2
=S
10
(S
30
−S
20
),
S
10
(S
30
−S
20
)=
1−r
a(1−r
10
)
(
1−r
a(1−r
30
)
−
1−r
a(1−r
20
)
)
=
1−r
a(1−r
10
)
(
1−r
a−ar
30
−a+ar
20
)
=
1−r
a(1−r
10
)
(
1−r
ar
20
(1−r
10
)
)
=a
2
r
20
(
1−r
1−r
10
)
2
This shows that LHS=RHS.
Hence proved.
Step-by-step explanation:
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