Question

Assertion (A): If \[ 2+6+16+40+\cdots \text{ to } k \text{ terms}=4608, \] then \(k=9\). Reason (R): \[ 2+3\cdot2+4\cdot2^2+\cdots+n\cdot2^{\,n-1}=n\cdot2^n, \qquad \forall n\in\mathbb N \] Which one of the following options is correct?

• (A) and (R) are true and (R) is the correct explanation of (A)
• (A) and (R) are true and (R) is not the correct explanation of (A)
• (A) is true but (R) is false
• (A) is false but (R) is true

Hint:

When the coefficients increase linearly and powers of 2 appear simultaneously, first identify the general term and then use the standard identity \(2+3\cdot2+4\cdot2^2+\cdots+n\cdot2^{n-1}=n2^n\).

Answer 1

The Correct Option is A