Sequences

Question

Okay, I will post without the answer so everyone can practice. (Will post answer later.) I apologize if I don't know the proper area of the forum for this, but I need help understanding how to set this up. So if you could all explain your logic, that would be great. 

In the arthimetic sequence p,r,s,t,u each term after the first is equal to the sum of the preceding term and a constant. Which of the following must also be an arthimetic 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

w07 · Answer

That is correct, I and II only.

Can you please explain your logic to help me understand?

Thanks so much.

Folaa3 · Answer

What i did was pick numbers randomly to test the consistency of each sequence. 

First scenario, let p, r, s, t u = 3, 5, 7, 9, 11 respectively. (Note that this follows the condition given in the stem that each term is the preceeding term plus a constant and in this case the constant = 2)

Test statement I : 2p,2r,2s,2t,2u = 6, 10, 14, 18, 22

Test statement II : p-3,r-3,s-3,t-3,u-3 = 0, 2, 4, 6, 8 

Test statement III: p^2,r^2,s^2,t^2,u^2 = 9, 25, 49, 81, 121

From the above, you can see that statements I and II are sequential and that statement III does not follow any sequence. 
Repeat the same for 2, 6, 10, 14, 18 (where constant = 4) and you will see that statement III fails to be an arithmetic sequence.

rthothad · Answer

Is there an easier way to do this, instead of picking numbers?

banerjeea_98 · Answer

Picking number in this case will waste lot of time. Picking number shud be the last resort.

Arithmetic seq means that the numbers shud be equally spaced i.e. the diff bet numbers in seq shud be the same.

We know r - p = c (constant)......we can write all the terms like this...so

1) 2(r-p) = 2c....all other terms have the same diff (2c)....so it is an arithmetic seq

2) r-3 - (p-3) = r - p = c....again an arithmetic seq

3) r^2 - p^2 = (r-p)(r+p)...we know r-p is = "c"...but don't know what r+p is.....so we can't say.

Hence I and II are the ans.

rthothad · Answer

Thanks baner. One more question.

I thought the sequence was p, (p+c), 2p+c, 4p+c.... doesn't the wording mean that, it says 'sum of the preceding term and a constant', in other words shouldn't the sequence be
2, 3(2+1), 6(2+3+1), 12..............

banerjeea_98 · Answer

No.....The statement
"each term after the first is equal to the  sum of the preceding term and a constant." 

means that the seq will be p, p+c, (p+c)+c, (p+2c)+c......

rthothad · Answer

OK. I read it as (sum of the preceding terms) and a constant.

Sequences

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