For The Infinite Sequence a1, a2, a3, ... an, an+1, an=3(an−1) For All GMAT Problem Solving

Question:For the infinite sequence, \(a_1,a_2,a_3,...a_n=3*a_{n-1}\) for all n>1. If \(a_5-a_2=156\), what is \(a_1\)?

1. -1
2. 2
3. 3
4. 4
5. 8

“For the infinite sequence a1, a2, a3, ... an, an+1, an=3(an−1) for all” - is a subject covered in the GMAT quantitative reasoning section. A student needs to be knowledgeable in a wide range of qualitative skills in order to successfully complete GMAT Problem Solving questions. There are 31 questions in the GMAT Quant section overall. Calculative mathematical problems must be solved in the GMAT quant topics' problem-solving section using appropriate mathematical skills.

Solutions and Explanation

Approach Solution : 1

Process elimination is a simpler approach to this problem's solution.
Starting with the first option
The sequences would be 1, 3, 5, 9, 27, and 81 if a1 = 1.
Here, a5 - a2 does not equal 156, so this is incorrect.
The second option
The sequences would be 2, 6, 18, 54, 162, 162 - 5 = 156 if a1 = 2.

This fits and is therefore the right answer.
Correct Answer: B

Approach Solution : 2

It is stated that a(n) = 3 x a. (n-1)

Thus:

a(2) = 3*a(1)
a(3) = 3*3*a(1) = 9a (1)
a(4) = 3*9*a(1) = 27a (1)
a(5) = 3*27*a(1) = 81a (1)
A(5) - A(2) is given to us as 156.

Thus:
81a(1) - 3a(1) = 156
78a(1) = 156
a(1) = 2
Correct Answer: B

Approach Solution : 3

\(a_n=3*a_{n-1}\)

Implies that, \(a_5-a_2=156\)

Implies that, \(a_5=27*a_2\)

Implies that, \(a_2=156/26=6=3*a_1\)

Implies that, \(a_1=2\)
Correct Answer: B

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