Question

p, r, s, t, u
An arithmetic sequence is a sequence in which each term after the first is equal to the sum of the preceding term and a constant. If the list of letters 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

Answer 1

The correct option is D I and II

$p,r,s,t,u$ are in AP,

$r-p=s-r=t-s=u-t$........... (1)

for case 1

$2r-2p=2(r-p)$

$2s-2r=2(s-r)$

$2t-2s=2(t-s)$

$2u-2t=2(u-t)$

from (1)

given series is AP

for case 2

$r-3-(p-3)=r-p$

$s-3-(r-3)=s-r$

$t-3-(s-3)=t-s$

$u-3-(t-3)=u-t$

from (1)

given series is AP

for case 3

$r^2-p^2=(r-p)(r+p)$

$s^2-r^2=(s-r)(s+r)$

$t^2-s^2=(t-s)(t+s)$

$u^2-t^2=(u-t)(u+t)$

given series is not in AP