find the sum to n terms: 1/1*4 + 1/4*7 + 1/7*10 ...
Answers
Answer:
n
3
n
+
1
.
Explanation:
Let,
t
n
denote the
n
t
h
term of the series,
s
n
=
1
1
⋅
4
+
1
4
⋅
7
+
1
7
⋅
10
+
...
to n terms.
Observe that, the First Factors of the Dr. of
t
n
are
1
,
4
,
7
,
...
,
which form an A.P., with the first term
a
=
1
,
and the
common difference,
d
=
3
;
∴
n
t
h
t
e
r
m
=
a
+
(
n
−
1
)
d
=
1
+
3
(
n
−
1
)
=
3
n
−
2
.
Similarly, for the Second Factors,
n
t
h
t
e
r
m
=
3
n
+
1
.
∴
t
n
=
1
(
3
n
−
2
)
(
3
n
+
1
)
.
∴
t
n
=
1
3
{
3
(
3
n
−
2
)
(
3
n
+
1
)
}
,
=
1
3
{
(
3
n
+
1
)
−
(
3
n
−
2
)
(
3
n
−
2
)
(
3
n
+
1
)
}
,
=
1
3
{
3
n
+
1
(
3
n
−
2
)
(
3
n
+
1
)
−
3
n
−
2
(
3
n
−
2
)
(
3
n
+
1
)
}
,
=
1
3
{
1
3
n
−
2
−
1
3
n
+
1
}
,
∴
s
n
=
1
3
n
∑
1
{
1
3
n
−
2
−
1
3
n
+
1
}
,
=
1
3
{
(
1
1
−
1
4
)
+
(
1
4
−
1
7
)
+
(
1
7
−
1
10
)
+
...
+
(
1
3
n
−
5
−
1
3
n
−
2
)
+
(
1
3
n
−
2
−
1
3
n
+
1
)
}
,
=
1
3
{
1
1
−
1
3
n
+
1
}
,
=
1
3
{
(
3
n
+
1
)
−
1
3
n
+
1
}
,
⇒
s
n
=
n
3
n
+
1