The correct option is D 2n+2n2+1⋅cosnπ4
We know that,
n∑r=0(−1)r nCr xr=(1−x)n
∴n∑r=0(−1)r nCr i5r=(1−i5)n=(1−i)n
Similarly,
n∑r=0(−1)r nCr i6r=(1−i6)n=(1−(−1))n
n∑r=0(−1)r nCr i7r=(1−i7)n=(1−(−i))n
n∑r=0(−1)r nCr i8r=(1−i8)n=(1−1)n
Hence,
n∑r=0(−1)r nCr⋅[i5r+i6r+i7r+i8r]=(1−i)n+(1−(−1))n+(1−(−i))n+(1−1)n
=(1−i)n+2n+(1+i)n+0=(√2)n(e−iπ/4)n+2n+(√2)n(eiπ/4)n=(√2)n(einπ/4+e−inπ/4)+2n=(√2)n(2cosnπ4)+2n=2n+2n2+1⋅cosnπ4
Alternate Solution
n∑r=0(−1)r nCr⋅[i5r+i6r+i7r+i8r]
Let n=1
=1∑r=0(−1)r 1Cr⋅[i5r+i6r+i7r+i8r]
=(−1)0 1C0⋅[i0+i0+i0+i0]+(−1)1 1C1⋅[i5+i6+i7+i8]
=4+(−1)[i−1−i+1]
=4+0=4
option 2n+2n2+1⋅cosnπ4 gives 4 at n=1