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Question

Sum of the series nr=0(1)r nCr[i5r+i6r+i7r+i8r], is

A
2n
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B
2n2+1
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C
nn+2n2+1
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D
2n+2n2+1cosnπ4
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Solution

The correct option is D 2n+2n2+1cosnπ4
We know that,
nr=0(1)r nCr xr=(1x)n

nr=0(1)r nCr i5r=(1i5)n=(1i)n

Similarly,
nr=0(1)r nCr i6r=(1i6)n=(1(1))n

nr=0(1)r nCr i7r=(1i7)n=(1(i))n

nr=0(1)r nCr i8r=(1i8)n=(11)n

Hence,
nr=0(1)r nCr[i5r+i6r+i7r+i8r]=(1i)n+(1(1))n+(1(i))n+(11)n
=(1i)n+2n+(1+i)n+0=(2)n(eiπ/4)n+2n+(2)n(eiπ/4)n=(2)n(einπ/4+einπ/4)+2n=(2)n(2cosnπ4)+2n=2n+2n2+1cosnπ4

Alternate Solution
nr=0(1)r nCr[i5r+i6r+i7r+i8r]
Let n=1
=1r=0(1)r 1Cr[i5r+i6r+i7r+i8r]
=(1)0 1C0[i0+i0+i0+i0]+(1)1 1C1[i5+i6+i7+i8]
=4+(1)[i1i+1]
=4+0=4
option 2n+2n2+1cosnπ4 gives 4 at n=1

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