COMEDK · Maths · 5. Sequences and Series
The sum of the series \((1+2)+\left(1+2+2^{2}\right)+\) \(\left(1+2+2^{2}+2^{3}\right)+\) upto \(n\) terms is
- A \(2^{n+2}-n-4\)
- B \(2\left(2^{n}-1\right)-n\)
- C \(2^{n+1}-n\)
- D \(2^{n+1}-1\)
Answer & Solution
Correct Answer
(A) \(2^{n+2}-n-4\)
Step-by-step Solution
Detailed explanation
The \(k\)-th term: \(T_k = 1 + 2 + 2^2 + \cdots + 2^k = 2^{k+1} - 1\) \(S_n = \displaystyle\sum_{k=1}^{n}(2^{k+1} - 1) = \sum_{k=1}^{n}2^{k+1} - n\) \(\displaystyle\sum_{k=1}^{n}2^{k+1} = 2^2 + 2^3 + \cdots + 2^{n+1} = 4 \cdot \dfrac{2^n - 1}{2-1} = 4(2^n-1) = 2^{n+2} - 4\)…
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