p, r, s, t, u An Arithmetic Sequence is a Sequence in Which Each Term GMAT Problem Solving

Question: 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?

2p, 2r, 2s, 2t, 2u
p-3, r-3, s-3, t-3, u-3
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

Correct Answer: D

Solution and Explanation
Approach Solution 1:

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.
This means the difference between each consecutive number is constant.

Let's look at the three choices provided in the options...

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

Since the difference is constant,say x and each number has been multiplied by a constant 2, the difference too will remain 2x. So it will be an arithmetic sequence...

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

Since the difference is constant,say again x and each number has been subtracted by a constant 3 , the difference too will remain x-3. So it will be an arithmetic sequence...

p^2,r^2,s^2,t^2,u^2

Since the difference is constant in initial sequence ,say x and now, each number has been multiplied by itself. Basically it means that each term is being multiplied by a different number, which is equal to itself. The difference now will change for each two consecutive numbers. So it will not be an arithmetic sequence.
If each number is multiplied by itself, the sequence will be 2*2, 3*3, 4*4. The difference between the results will vary- 4, 9, 16. Hence it is not an arithmetic sequence.

As per the above explanation, the correct option is D since I and II are arithmetic sequences.

Approach Solution 2:

The defining factor of an arithmetic sequence is that there is a constant difference d between each pair of successive terms.

We are given an arithmetic sequence p, r, s, t, u. We need to determine which of the following MUST also be an arithmetic sequence. An easy way to determine this will be to choose convenient numbers for our initial sequence.

Let's let the sequence look like this:

p, r, s, t, u = 2, 4, 6, 8, 10.

Notice that the constant difference between each pair of successive terms is d = 2, and thus we are assured that it is an arithmetic sequence.

We can now use these numbers in the sequences presented in the three statements.

Statement I: 2p, 2r, 2s, 2t, 2u → (2 x 2), (2 x 4), (2 x 6), (2 x 8), (2 x 10) →

4, 8, 12, 16, 20

Notice that the above number set follows the definition of an arithmetic sequence, with a constant difference of d = 4. Thus, Statement I must be true.

We can eliminate answer choices B, C, and E.

Statement II. (p – 3), (r – 3), (s – 3), (t – 3), (u –3) →

(2 – 3), (4 – 3), (6 – 3), (8 – 3), (10 – 3) →

-1, 1, 3, 5, 7

Notice that the above number set follows the definition of an arithmetic sequence, with a constant difference of d = 2. Thus, Statement II must be true.

We can eliminate answer choice A. Even though we know that D is the correct answer choice, let’s check statement III anyway.

Statement III. p^2, r^2, s^2, t^2, u^2 →

2^2, 4^2, 6^2, 8^2, 10^2 →

4, 16, 36, 64, 100

Notice that the above number set DOES NOT follow the definition of an arithmetic sequence because there is not a constant difference between each pair of successive terms in the set. Thus, Statement III is NOT true.

Thus the correct answer is D

Approach Solution 3:
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
r2−p2=(r−p)(r+p)

s2−r2=(s−r)(s+r)

t2−s2=(t−s)(t+s)

u2−t2=(u−t)(u+t)

given series is not in AP

“p, r, s, t, u An arithmetic sequence is a sequence in which each term”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Prep Plus 2020". To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.

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