in an arthimetic sequence the sum of 15 term is equal to that of 25 term find its sum of first 40 term
Answers
Consider the following example:
The first term of an arithmetic sequence is
2
and the third is
6
. What is
d
, the common difference?
With an arithmetic sequence, the
d
is added to each term to get the next.
Since
t
1
=
2
and
t
3
=
6
, there will be
3
−
1
=
2
d
'
s
added to
t
1
to get
t
3
. So, we can write the following equation:
2
+
2
d
=
6
2
d
=
4
d
=
2
It works, too, since if
t
1
=
2
,
t
2
=
4
and
t
3
=
6
, which makes an arithmetic sequence.
The same principle can be applied to our problem.
25
−
1
=
24
, so there will be
24
d
'
s
added to
51
to get
99
.
Hence,
51
+
24
d
=
99
24
d
=
48
d
=
2
So, the common difference is
2
.
All we have to do now is to apply the formula
s
n
=
n
2
(
2
a
+
(
n
−
1
)
d
)
)
to determine the sum of the sequence.
s
15
=
15
2
(
2
(
51
)
+
(
15
−
1
)
2
)
s
15
=
15
2
(
102
+
28
)
s
15
=
15
2
(
130
)
s
15
=
975
Thus, the sum of the first fifteen terms in the arithmetic sequence is
975
.
Hopefully this helps!