AP EAMCET201824 Apr 2018Morning ShiftMathematicsBinomial TheoremActual
If C_r= ^n C_r , then C₀+C₄+C₈+C₁₂+ =
Options
- A2^ n 2 [ n 4 +2^ n 2 -1 ] 2
- B2^ n 2 n 4
- C2^ n-1 n 4
- D2^ n 2 [ n 4 +2^ n 2 -1 ] 2
Correct answer
D. 2^ n 2 [ n 4 +2^ n 2 -1 ] 2
Step-by-step solution
In given series difference of suffies is 4 0-4=8-4=12-4= =4 Now, ( l )^ 1 4 =( 0+i 0)^ 1 4 aligned & =( 2 r +i 2 r )^ 1 4 & = r 2 +i r 2 aligned where r=0,1,2,3 Four roots of unity are 1, i,-1,-i aligned & =1, , , ^2 say and (1+x)^n & = _ r=0 ^n ^n C_r x^r aligned For r=0,4,8,12 R.H.S of Eq. (v) array r ^n C₀(1+1+1+1)+ ^n C₄ (1+ ^4+ ^8+ ¹² ) + ^n C₈ (1+ ^8+ ¹⁶+ ²⁴ )+ + =4 ( ^n C₀+ ^n C₄+ ^n C₈ ) ( ^4=1 ) . array L.H.S of Eq. (v) aligned & =2^n+(1+i)^n+ (1+i^2 )^n+ (1+i^3 )^n & =2^n+(1+i)^n+0+(1-i)^n aligned Since,