AP EAMCET202317 May 2023Evening ShiftMathematicsBinomial TheoremActual
If C _ j stands for ^ n C _ j , then C ₀ 2 + C ₁ 2.2^2 + C ₂ 3.2^3 + + C _ n ( n +1) 2^ n +1 =
Options
- A3^n 2^ n+1 (n+1)
- B3^ n+1 2^ n+1 (n+1)
- C3^ n 2^ n ( n +1)
- D3^ n +1 2^ n ( n +1)
Correct answer
B. 3^ n+1 2^ n+1 (n+1)
Step-by-step solution
C₀ 2 + C₁ 2.2^2 + C₂ 3.2^3 + + C_n (n+1) 2^ n+1 aligned & = _ r=0 ^n ^n C_r r+1 ( 1 2 )^ r+1 & = 1 n+1 _ r=0 ^n n+1 r+1 ^n C _r ( 1 2 )^ r+1 = 1 n+1 _ r=0 ^n ^ n+1 C _ r+1 ( 1 2 )^ r+1 & 1 n+1 (1+ 1 2 )^ n+1 = 3^ n+1 ( n +1) 2^ n +1 aligned