GATE CS 2015 Set 2 — Question 9

Let the arithmetic sequence be $a_1, a_2, a_3, a_4, a_5$ with common difference $d$. So, $P=a_1, R=a_1+d, S=a_1+2d, T=a_1+3d, U=a_1+4d$.Statement I: $2P, 2R, 2S, 2T, 2U$
The terms are $2a_1, 2(a_1+d), 2(a_1+2d), 2(a_1+3d), 2(a_1+4d)$.
This can be written as $2a_1, 2a_1+2d, 2a_1+4d, 2a_1+6d, 2a_1+8d$.
This is an arithmetic sequence with the first term $2a_1$ and a common difference of $2d$.
So, Statement I is TRUE.Statement II: $P-3, R-3, S-3, T-3, U-3$
The terms are $a_1-3, (a_1+d)-3, (a_1+2d)-3, (a_1+3d)-3, (a_1+4d)-3$.
This can be written as $(a_1-3), (a_1-3)+d, (a_1-3)+2d, (a_1-3)+3d, (a_1-3)+4d$.
This is an arithmetic sequence with the first term $a_1-3$ and a common difference of $d$.
So, Statement II is TRUE.Statement III: $P^2, R^2, S^2, T^2, U^2$
The terms are $a_1^2, (a_1+d)^2, (a_1+2d)^2, (a_1+3d)^2, (a_1+4d)^2$.
Let's check the differences between consecutive terms:
Difference between the first two terms: $(a_1+d)^2 - a_1^2 = a_1^2 + 2a_1d + d^2 - a_1^2 = 2a_1d + d^2$.
Difference between the second and third terms: $(a_1+2d)^2 - (a_1+d)^2 = (a_1^2 + 4a_1d + 4d^2) - (a_1^2 + 2a_1d + d^2) = 2a_1d + 3d^2$.
For the sequence to be an arithmetic sequence, these differences must be equal:
$2a_1d + d^2 = 2a_1d + 3d^2$
$d^2 = 3d^2$
$2d^2 = 0$
This implies $d=0$. If $d=0$, then all terms $P, R, S, T, U$ are identical, and their squares would also form an arithmetic sequence (with common difference 0). However, for a general arithmetic sequence where $d \neq 0$, the squares do not form an arithmetic sequence. For example, if $P=1, R=2, S=3$ (common difference $d=1$), then $P^2=1, R^2=4, S^2=9$. The differences are $4-1=3$ and $9-4=5$, which are not equal. Thus, Statement III is generally FALSE.Based on the analysis, only statements I and II are true.The final answer is $\boxed{\text{B}}$