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Question
Mathematics
If the sum of the series 1+(2/3)+(6/9)+(10/27)+(14/81)+ ldots is 2 m-5, then find the value of m.
Q. If the sum of the series
1
+
3
2
+
9
6
+
27
10
+
81
14
+
…
is
2
m
−
5
, then find the value of
m
.
107
159
Sequences and Series
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Answer:
4
Solution:
Let
S
=
1
+
3
2
+
9
6
+
27
10
+
81
14
+
…
⇒
S
=
1
+
3
2
+
3
2
6
+
3
3
10
+
3
4
14
+
…
⇒
S
−
1
=
3
2
+
3
2
6
+
3
3
10
+
3
4
14
+
…
...
(
i
)
⇒
3
1
(
S
−
1
)
=
3
2
2
+
3
3
6
+
3
4
10
+
…
...
(
ii
)
Subtracting (ii) from (i), we get
3
2
(
S
−
1
)
=
3
2
+
3
2
4
+
3
3
4
+
3
4
4
+
…
⇒
3
2
(
S
−
1
)
=
3
2
+
1
−
3
1
9
4
⇒
3
2
(
S
−
1
)
=
3
4
⇒
S
=
3
⇒
2
m
−
5
=
3
⇒
m
=
4