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Byju's Answer
Standard X
Mathematics
Sum of N Terms of an AP
If the sum of...
Question
If the sum of
m
terms of an A.P. is the same as the sum of its
n
terms, show that the sum of its
(
m
+
n
)
terms is zero.
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Solution
Let
a
be the first term and
d
be the common difference of the given A.P. Then,
S
m
=
S
n
.
⟹
m
2
{
2
a
+
(
m
−
1
)
d
}
=
n
2
{
2
a
+
(
n
−
1
)
d
}
⟹
2
a
(
m
−
n
)
+
{
m
(
m
−
1
)
−
n
(
n
−
1
)
}
d
=
0
⟹
2
a
(
m
−
n
)
+
{
(
m
2
−
n
2
)
−
(
m
−
n
)
}
d
=
0
⟹
(
m
−
n
)
{
2
a
+
(
m
+
n
−
1
)
d
}
=
0
⟹
2
a
+
(
m
+
n
−
1
)
d
=
0
[
∵
m
−
n
≠
0
] ...(i)
Now,
S
m
+
n
=
m
+
n
2
{
2
a
+
(
m
+
n
−
1
)
d
}
⟹
S
m
+
n
=
m
+
n
2
×
0
=
0
[Using (i)]
Suggest Corrections
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