An arithmetic sequence is a sequence in which each term

D is the answer

a(n) = a(n-1) + d

1) b(n) = 2*a(n) = 2*a(n-1)+2*d = b(n-1)+2*d -> arithmetic (d_new = 2*d)
2) b(n) = a(n) - c = a(n-1)+d-c = a(n-1)-c+d = b(n-1) + d -> arithmetic
3) b(n) = a(n)^2 = a(n-1)^2 + 2*a(n-1)*d + d^2 = b(n-1) + 2*a(n-1)*d + d^2 <> b(n-1) + const -> not arithmetic

From the definition you can clearly see that II is an arithmetic sequence. III is clearly not.

Option I is a trap! If you expand it to \(p+p, r+r, s+s, t+t, u+u\) it appears it doesn't increase by a constant.

If you rewrite the original sequence with an increase of \(n\) you get, \(p, p + n, p + 2n, p+3n, p+4n\).

Now multiply it by 2, and get \(2p, 2p + 2n, 2p + 4n, 2p + 6n, 2p + 8n\). Now you can see that I increases by \(2n\)!

1) multiples all the terms by 2
2) subtracts all the terms by 3..

They gotto be in arithmetic sequence

whereas 3 squares .. anythign when squared will increase the value by a huge value. . and arithmetic sequence may not be maintained

I followed a mixed approach. Started by picking numbers and realized soon that option 3 would not give me the results

Hi,

Can someone please clarify a nagging issue:

For a sequence to be an arithmetic sequence, does the different between the units have to be constant? To elaborate, are all three examples below considered arithmetic sequences?

-[2,4,6,8] = difference of 2
-[3,9,81...] = all the units are squared but the differences are not constant
-[2, 5, 9, 14]= the difference is 2+1 so (5= 2 + 3), (9=3+4), (14 = 9 + 5)

Arithmetic Progression is a special type of sequence in which the difference between successive terms is constant.

{2, 4, 6, 8} is an arithmetic progression (the difference = 2).
{3, 9, 81} is neither arithmetic not geometric progression.
{2, 5, 9, 14} is neither arithmetic not geometric progression.

Best thing should be plug in:

p = 1, q = 2, r = 3, s = 4, t = 5

I. 2, 4, 6, 8, 10 (Arithmetic Progression)
II. -2, -1, 0, 1, 2 (Arithmetic Progression)
III. 1, 4, 9, 16, 25 (Not AP)

Hence I and II

hi mawus,

as we know 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....
lets look at the three choices...
I. 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...

II. 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...

III. 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 number ... so it will not be an arithmetic sequence...

only l and ll.. ans D....

The list of letters is bugging me, specifically because there is a gap between p and r (q). Is this relevant or just a distraction in the question? A simple explanation would be very helpful! Thank you!

Valid question, but it is an official question. It is what it is and you can not question the language or the OA for the question.

I dont believe knowing the alphabets in a particular sequence is relevant for this question. GMAT can even give us strange symbols to stand for these variables. You just have to understand what is an arithmetic progression and after that it is all a matter solving the question with whatever variables are given to you.

If GMAT wants to give you a,z,y,b,c and say that they are in arithmetic progression (ie difference between 2 consecutive terms is constant and is the same value), then you have to stick to this pattern of variables.

Hope this helps.

Please tag Sequence as this is Arithmetic Series question.
Thank you.

Attached is a visual that should help.

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.

Answer D

Note: Note that in the answer choices presented, the option “I, II, and III” is not given. Thus, once we determined that I and II were true we could have immediately chosen D as our answer.

If you try to verify I and II, you'll see they're valid sequence. As there is no answer option like I, II and III , you should not try to verify III (as you already know that I, II are valid sequence). Verifying III (at least here) is just waste of time.
Thanks__

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