If (a1 + a2 + a3 + .... +an) = 3(2n+1 - 2), For Every n≥1, Then a11 Equals

Question: If \((a1 + a2 + a3 + .... +an) = 3(2^{n+1} - 2)\), for every n≥1, then a11 equals

1. 1536
2. 2012
3. 2048
4. 3072
5. 6144

“If (a1 + a2 + a3 + .... +an) = 3(2n+1 - 2), For Every n≥1, Then a11 Equals”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Official Guide Quantitative Review".  To solve GMAT Problem Solving questions a student must have knowledge about a good number of qualitative skills. GMAT Quant section consists of 31 questions in total. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1:

\(If (a1 + a2 + a3 + .... +an) = 3(2^{n+1} - 2)\), for every n≥1, then a11 equals

If n=10, then
—> a1 + ... + a10 = 3(\(2^{11}\) - 2)

If n=11, then
—> a1+... + a10 + a11= 3(\(2^{12}\) - 2)

Well, subtracting 2nd(n=11) statement from 1st (n=10):

—>\( a11 = 3(2^{12} - 2) - 3(2^{11} - 2) =\)

\(3(2^{12} - 2^{11}) = 3*2^{11} (1) = 3* 2048 =614\)4

Answer (E)

Approach Solution 2:

\((a1 + a2 + a3 + .... +an) = 3(2^{n+1} - 2)\)

Substitute, n = 10

--> \((a1 + a2 + a3 + .... +a10) = 3(2^{10+1} - 2)\)

--> \((a1 + a2 + a3 + .... +a10) = 3(2^{11} - 2)\) — (1)

Substitute, n = 11

--> \((a1 + a2 + a3 + .... +a11) = 3(2^{11+1} - 2)\)

--> \((a1 + a2 + a3 + .... +a11) = 3(2^{12} - 2)\)

(1) - (2) gives,
--> (a1+a2+a3+…+a10+a11) - (a1+a2+a3+…+a10) = \(3(2^{12} - 2) - 3(2^{11} - 2)\)(1) − (2) ........ (2)

--> a11= 3\((2^{12} - 2^{11}) = 3*2^{11}\) (1) = 3* 2048 =6144

The correct answer is (E).

Approach Solution 3:

a1+a2+a3+…+an = \(3(2^{n+1} - 2)\), for every n ≥ 1,

or, a1 + a2 + a3 +…+ an= \(3(2^{n+1} - 2)\)

a1 = \(3(2^{1+1} - 2)\)

a1+a2 = \(3(2^{2+1} - 2)\)

=> a2 = 3\((2^3 - 2^2)\)

therefore, a11 = \(3(2^{12} - 2^{11})\)

or, a11 = 6 *1024 = 6144

correct answer is E

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