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UPSC CDS Math Square, Roots and Surds Questions
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© examsnet.com
Question : 15
Total: 22
What is
√
1
+
1
1
2
+
1
2
2
+
√
1
+
1
2
2
+
1
3
2
+
.
.
.
+
√
1
+
1
2007
2
+
1
2008
2
equal to ?
[2016 CDS-II]
2008
−
1
2008
2007
−
1
2007
2007
−
1
2007
2008
−
1
2009
Validate
Solution:
t
r
=
√
1
+
1
r
2
+
1
(
r
+
1
)
2
=
√
r
2
(
r
+
1
)
2
+
(
r
+
1
)
2
+
r
2
r
2
(
r
+
1
)
2
=
√
r
2
(
r
+
1
)
2
+
(
r
+
(
r
+
1
)
×
1
+
1
r
2
(
r
+
1
)
2
=
r
(
r
+
1
)
+
1
r
(
r
+
1
)
=
1
+
1
r
(
r
+
1
)
=
1
+
1
r
−
1
r
+
1
∴
√
1
+
1
1
2
+
1
2
2
+
√
1
+
1
2
2
+
1
3
2
+
.
.
.
+
√
1
+
1
2007
2
+
1
2008
2
=
2007
∑
i
=
1
t
r
=
2007
∑
i
=
1
(
1
+
1
r
−
1
r
+
1
)
=
(
1
+
1
1
−
1
2
)
+
(
1
+
1
2
−
1
3
)
+
(
1
+
1
3
−
1
4
)
+
.
.
.
+
(
1
+
1
2006
−
1
2007
)
+
(
1
+
1
2007
−
1
2008
)
=
2008
−
1
2008
© examsnet.com
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