AP EAMCET · Maths · Binomial Theorem
If \((1+x)^n=\sum_{r=0}^n C_r x^r\), then the value of \(C_0+\left(C_0+C_1\right)+\left(C_0+C_1+C_2\right)+\ldots+\) \(\left(\mathrm{C}_0+\mathrm{C}_1+\mathrm{C}_2+\ldots+\mathrm{C}_{\mathrm{n}}\right)\) is
- A \(\mathrm{n} 2^{\mathrm{n}-1}\)
- B \(2^{\mathrm{n}}+\mathrm{n}\)
- C \((\mathrm{n}+2) 2^{\mathrm{n}}\)
- D \((\mathrm{n}+2) 2^{\mathrm{n}-1}\)
Answer & Solution
Correct Answer
(D) \((\mathrm{n}+2) 2^{\mathrm{n}-1}\)
Step-by-step Solution
Detailed explanation
\(S = \sum_{r=0}^n (n-r+1)C_r\) \(S = (n+1)\sum_{r=0}^n C_r - \sum_{r=0}^n r C_r\) \(S = (n+1)2^n - n2^{n-1}\) \(S = (2(n+1) - n)2^{n-1}\) \(S = (2n+2-n)2^{n-1}\) \(S = (n+2)2^{n-1}\)
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