if the sum of first 6 terms of any GP is equal to 9 times the sum of the first three terms then find the common ratio of the GP
Answers
Consider a GP as
a
,
a
r
,
a
r
2
,
a
r
3
,
a
r
4
,
a
r
5
Sum of first 3 terms
=
a
+
a
r
+
a
r
2
⋯
(
1
)
Sum of first 6 terms
=
a
+
a
r
+
a
r
2
+
a
r
3
+
a
r
4
+
a
r
5
=
a
+
a
r
+
a
r
2
+
r
3
(
a
+
a
r
+
a
r
2
)
⋯
(
2
)
Let
a
+
a
r
+
a
r
2
=
x
.
Then,
Sum of first 3 terms
=
x
Sum of first 6 terms
=
x
+
r
3
x
=
x
(
r
3
+
1
)
x
(
r
3
+
1
)
:
x
=
9
:
1
(
r
3
+
1
)
:
1
=
9
:
1
r
3
+
1
=
9
r
3
=
8
r
=
2
common ratio = 2
Solution 2
a
(
r
6
−
1
)
r
−
1
:
a
(
r
3
−
1
)
r
−
1
=
9
:
1
(
r
6
−
1
)
:
(
r
3
−
1
)
=
9
:
1
(
r
6
−
1
)
=
9
(
r
3
−
1
)
r
6
−
1
=
9
r
3
−
9
r
6
−
9
r
3
+
8
=
0
⋯
(
1
)
Let
x
=
r
3
. Then, (1) becomes
x
2
−
9
x
+
8
=
0
(
x
−
8
)
(
x
−
1
)
=
0
x
=
8
or
1
Since
r
=
3
√
x
,
r
can be 2 or 1
But
r
=
1
may not be a solution because there was a term
(
r
−
1
)
in the denominator in the initial formula which would be zero when
r
=
1
and division by zero is not defined. Consider a GP with
r
=
1
as "3, 3, 3, 3, 3, 3". We can see that the given condition is not satisfied because the ratio is not 9. Hence
r
=
1
is not a solution.