GATE CS 2015 Set 2 QGA-6 — General Aptitude

Let the given sequence $P, R, S, T, U$ be an arithmetic progression (AP) with a common difference $d$. Thus, we can write:

• $R = P + d$
• $S = P + 2d$
• $T = P + 3d$
• $U = P + 4d$

Let's evaluate each statement:

1. Statement I: $2P, 2R, 2S, 2T, 2U$
   The difference between consecutive terms is: $$(2R) - (2P) = 2(R - P) = 2d$$ $$(2S) - (2R) = 2(S - R) = 2d$$ Since the difference between any two consecutive terms is constant ($2d$), this sequence is an arithmetic sequence.
   (Statement I is correct)


2. Statement II: $P-3, R-3, S-3, T-3, U-3$
   The difference between consecutive terms is: $$(R - 3) - (P - 3) = R - P = d$$ $$(S - 3) - (R - 3) = S - R = d$$ Since the difference between any two consecutive terms is constant ($d$), this sequence is also an arithmetic sequence.
   (Statement II is correct)


3. Statement III: $P^2, R^2, S^2, T^2, U^2$
   Let us test this with a simple counterexample. Let the AP be $1, 2, 3, 4, 5$ (where $d = 1$).
   Squaring the terms gives: $1, 4, 9, 16, 25$.
   The differences between consecutive terms are: $$4 - 1 = 3$$ $$9 - 4 = 5$$ Since the common difference is not constant, this is not an arithmetic sequence.
   (Statement III is incorrect)

Hence, only I and II are in an arithmetic sequence.