the medians of first 15 terms of two arithmatic sequence with same common difference are 25 and 28, find the sum of first 15 terms of these two sequences
Answers
Answer:
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
.