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KVPY SA Exam 2014 Question Paper
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© examsnet.com
Question : 3
Total: 80
The sum of all positive integers n for which
1
3
+
2
3
+
.
.
.
.
+
(
2
n
)
3
1
2
+
2
2
+
.
.
.
.
+
n
2
is also an integer is
8
9
15
Infinite
Validate
Solution:
(
2
n
(
2
n
+
1
)
2
)
2
n
(
n
+
1
)
(
2
n
+
1
)
6
=
6
n
2
(
2
n
+
1
)
2
n
(
n
+
1
)
(
2
n
+
1
)
6
n
(
2
n
+
1
)
(
n
+
1
)
=
6
n
(
n
+
1
+
n
)
n
+
1
=
6
n
+
6
n
2
n
+
1
=
6
n
+
6
(
n
−
1
)
+
6
n
+
1
=
Integer
The values of 'n' which are satisfying are n = 1, 2, 5 only for being integer
∴ sum = 1 + 2 + 5 = 8
© examsnet.com
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