The Sequence \(a_1,a_2,...,a_n,...\)is such that \(a_n=2a_{n-1}-x\) GMAT Problem Solving

Question: The sequence \(a_1,a_2,...,a_n,...\)is such that \(a_n=2a_{n-1}-x\)for all positive integers \(n\geq2\)and for certain number x. If \(a_5=99,a_3=27\), what is the value of x?

1. 3
2. 9
3. 18
4. 36
5. 45

Answer:
Solution with Explanation:
Approach Solution (1):

We know that:

\(a_5=2*a_4-x=99\)
\(a_4=2*a_3-x=2*27-x\)

Therefore,

\(a_5=2*(54-x)-x=99\)
\(108-3*x=99\)

Therefore, x = 3

Correct Option: A

Approach Solution (2):

Given:

\(a_n=2a_{n-1}-x\)
\(a_5=99\)
\(a_3=27\)
\(a_5=2a_4-x=2(a_3-x)-x=4a_3-3x=99\)
\(4(27)-3x=993x=108-99=9\)
\(x=3\)

Correct Option: A

Approach Solution (3):

Plug the known values \(a_5=99,a_3=27\)into the formula:

\(a_5=2(a_4)-x\)
\(99=2(a_4)-x\)
\(a_4=2(a_3)-x\)
\(a_4=2(27)-x=54-x\)
Substitute 54 – x for \(a_4\) in the top equation:
99 = 2(54 – x) – x
99 = 108 – 3x
3x = 9
x = 3
Correct Option: A

“The sequence \(a_1,a_2,...,a_n,...\)is such that \(a_n=2a_{n-1}-x\) for all positive integers \(n\geq2\) and for certain number x. If \(a_5=99,a_3=27\), what is the value of x?”- 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 amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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