GATE CS 2015 Set 2 Q.6 (GA Q.6): Quantitative Aptitude

If the list of letters, $P, R, S, T, U$ is an arithmetic sequence, which of the following are also in arithmetic sequence?

I. $2P, 2R, 2S, 2T, 2U$
II. $P - 3, R - 3, S - 3, T - 3, U - 3$
III. $P^2, R^2, S^2, T^2, U^2$

I only

I and II

II and III

I and III

Official answer and solution

Official answer: B
From: Official GATE 2015 answer key

Solution

Apply linear algebraic transformations to the terms of an arithmetic progression.

Let $P, R, S, T, U$ be in arithmetic sequence with common difference $d$ :

R - P = S - R = T - S = U - T = d.

Examine Statement I: $2P, 2R, 2S, 2T, 2U$ .
Difference between consecutive terms:

2R - 2P = 2(R - P) = 2d.

2S - 2R = 2(S - R) = 2d.

The difference is constant ( $2d$ ), so Statement I forms an arithmetic sequence.

Examine Statement II: $P - 3, R - 3, S - 3, T - 3, U - 3$ .
Difference between consecutive terms:

(R - 3) - (P - 3) = R - P = d.

The difference remains constant ( $d$ ), so Statement II forms an arithmetic sequence.

Examine Statement III: $P^2, R^2, S^2, T^2, U^2$ .
Consider $P = 1, R = 2, S = 3, T = 4, U = 5$ with $d = 1$ .
Squaring terms yields $1, 4, 9, 16, 25$ .
The differences are $4 - 1 = 3$ and $9 - 4 = 5$ , which are not equal.
Hence, Statement III is not in arithmetic sequence in general.

Therefore, both I and II are in arithmetic sequence.

Final answer

Common trap

Assuming squares of terms in an AP also form an AP.

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2015 Computer Science and Information Technology (CS) Session 2
Question: 6 of 65, printed as GA Q.6
Section: General Aptitude

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