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Question

If Sn=cot1(3)+cot1(7)+cot1(13)+cot1(21)+n terms, then

A
S10=tan156
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B
S=π4
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C
S6=sin135
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D
S20=cot1(1.1)
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Solution

The correct options are
A S10=tan156
B S=π4
C S6=sin135
D S20=cot1(1.1)
Let tr denote the rth term of the series 3,7,13,21,... and
S= 3+7+13+21+...+tn
S=0371321...tn1tn
__________________________________
0=3+4+6+8+...+2ntn
tn=3+4+6+...+2n=1+2×12n(n+1)
=n2+n+1
Let Tr=cot1(r2+r+1)
=tan1(1r2+r+1)

=tan1(r+1r1+r(r+1))
=tan1(r+1)tan1r
Thus, the sum of first n terms of the given series is
nr=1[tan1(r+1)tan1r]
=tan1(n+1)tan1(1)
=tan1[n+111+1(n+1)]=tan1(nn+2)
=tan1⎜ ⎜ ⎜11+2n⎟ ⎟ ⎟
S=limntan1⎜ ⎜ ⎜11+2n⎟ ⎟ ⎟=π4,

S10=tan11012=tan156

S6=tan134=sin135

S20=tan11011=cot11.1

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