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

Concept: The given series is of the form \[ 2+3\cdot2+4\cdot2^2+\cdots \] whose \(n\)-th term is \[ (n+1)2^{n-1}. \] We use the identity \[ 2+3\cdot2+4\cdot2^2+\cdots+n\cdot2^{n-1}=n\cdot2^n. \]

Step 1: Express the given series in standard form. The given series is \[ 2+6+16+40+\cdots \] Its terms can be written as \[ 2=2\cdot2^0,\quad 6=3\cdot2^1,\quad 16=4\cdot2^2,\quad 40=5\cdot2^3. \] Therefore, the sum of \(k\) terms is \[ S_k=2+3\cdot2+4\cdot2^2+\cdots+(k+1)2^{k-1}. \]

Step 2: Apply the given identity. Replacing \(n\) by \(k+1\) in the identity, \[ S_k=(k+1)2^{k+1}. \] Given \[ S_k=4608, \] so \[ (k+1)2^{k+1}=4608. \]

Step 3: Determine the value of \(k\). Since \[ 4608=9\times512=9\times2^9, \] we get \[ (k+1)2^{k+1}=9\cdot2^9. \] Comparing factors, \[ k+1=9 \] and \[ 2^{k+1}=2^9. \] Hence, \[ k=8. \] Since the series starts with the term corresponding to \(n=2\), the total number of terms is \[ 9. \] Therefore, \[ k=9. \] Hence, Assertion (A) is true.

Step 4: Examine the Reason. The identity \[ 2+3\cdot2+4\cdot2^2+\cdots+n\cdot2^{n-1}=n\cdot2^n \] is true and is exactly the result used to evaluate the sum. Therefore, Reason (R) is true and correctly explains Assertion (A). \[ \boxed{\text{(A) and (R) are true and (R) is the correct explanation of (A)}} \]