Problem Solving

Hi shahfahad,

I'm going to give you a couple of hints so that you can re-attempt this question on your own:

1) This question can be solved by TESTing VALUES.
2) An arithmetic sequence is one that increases (or decreases) by a consistent 'number' as you go from term to term.

Some arithmetic sequences include....
1, 2, 3, 4, 5...
3, 2, 1, 0, -1, -2...
-1, -0.5, 0, 0.5, 1, 1.5, 2, 2.5...

With the above information, could you determine which of the 3 sequences in this question are ARITHMETIC sequences?

GMAT assassins aren't born, they're made,
Rich

shahfahad wrote: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

Please could you explain how to make the arithmetic sequence.

Basically, an arithmetic sequence is a sequence in which each term can be calculated by adding some constant, k, to the preceding term.

Some examples:
6, 8, 10, 12, 14,... (adding 2 to each term to get the next term)
-1, 6, 13, 20, 27,... (adding 7 to each term to get the next term)
10, 7, 4, 1, -2, -5,.... (adding -3 to each term to get the next term)

-----------------------------------------

We're told that p,r,s,t,u is an arithmetic sequence, so let's say that each term is derived by adding k to the previous term.
In other words, r - p = k, and s - r = k, and t - s = k and u - t = k

Now let's check the options:

I. 2p,2r,2s,2t,2u
Is it the case that each term is derived by adding SOME CONSTANT to the previous term?
Yes!
Observe that 2r - 2p = 2(r - p) = 2k
Likewise, 2s - 2r = 2(s - r) = 2k
And 2t - 2s = 2(t - s) = 2k
And so on.
Since each term is derived by adding 2k to the previous term, this is an ARITHMETIC SEQUENCE


II. p-3, r-3, s-3, t-3, u-3
Is it the case that each term is derived by adding SOME CONSTANT to the previous term?
Yes!
Observe that (r-3) - (p-3) = (r - p) = k
Likewise, (s-3) - (r-3) = (s - r) = k
And so on.
Since each term is derived by adding k to the previous term, this is an ARITHMETIC SEQUENCE

NOTE: At this point, we can stop, because only one answer choice is valid if sequences I and II are arithmetic

Answer: D

Cheers,
Brent

shahfahad wrote: Each equation equals to 2k. If one of it equals 2k and the next sequence equals k. Then it would not be arithmetic?

If we derive one term by adding 2k to the previous term, and we derive another term by adding k to the previous term, then the sequence isn't guaranteed to be arithmetic. The only way this COULD be an arithmetic sequence is if k = 0. However, the question asks "which of the following must also be an arithmetic sequence?"

In other words, it must be guaranteed to be an arithmetic sequence

Cheers,
Brent