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

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

A. I only
B. I and II
C. II and III
D. I and III

Accepted Answer

Only Sequences I and II are arithmetic progressions. Answer: B. I and II. A sequence is arithmetic if the difference between consecutive terms is constant. If P, R, S, T, U form an AP with common difference d, we must check which transformations preserve this constant-difference property. Multiplying every term by the same constant preserves the ratio of differences, so 2P, 2R, ... is still an AP. Subtracting the same constant from every term shifts all terms equally, leaving the differences unchanged, so P-3, R-3, ... is also an AP. Squaring, however, changes the differences: the difference between consecutive squares depends on the actual values, not just d, so P^2, R^2, ... is generally NOT an AP.

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 A. I only B. I and II C. II and III D. I and III