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## GMAT - Problem Solving - Sequences and series

### An arithmetic sequence is a sequence in which each term after the first term is equal to the sum of the preceding term and a constant. If the list of numbers shown above i (more)

p, r, s, t, u

An arithmetic sequence is a sequence in which each term after the first term is equal to the sum of the preceding term and a constant. If the list of numbers shown above is an arithmetic sequence, which of the following must also be an 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) II only
(C) III only
(D) I and II
(E) II and III

given p, r, s, t u

are in AP

assume common difference be d

i.e.

r = p + d, s = p + 2d, t = p + 3d , u = p + 4d

I. 2p, 2r, 2s, 2t, 2u

2r = 2(p + d) = 2p + 2d

2s = 2(p + 2d) = 2p + 4d

2t = 2(p + 3d) = 2p + 6d

2u = 2(p + 4d) = 2p + 8d

they are in AP, with first term 2p and common difference 2d

II. p-3, r-3, s-3, t-3, u-3

r - 3 = p + d -3

s - 3 = p + 2d - 3

t - 3 = p + 3d - 3

u - 3 = p + 4d - 3

The sequence is in AP with first term p-3 and common diffrence d

We don't need to check third option as only Option D contains both I and II