BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem

An arithmetic sequence

This topic has expert replies
Post new topic Post Reply
Previous Topic Next Topic
Senior | Next Rank: 100 Posts
Posts: 42
Joined: Fri Oct 30, 2015 9:01 am

An arithmetic sequence

by shahfahad » Thu Nov 26, 2015 2:07 am
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.
Top
Source: — Problem Solving |

GMAT/MBA Expert

User avatar
Elite Legendary Member
Posts: 10392
Joined: Sun Jun 23, 2013 6:38 pm
Location: Palo Alto, CA
Thanked: 2867 times
Followed by:511 members
GMAT Score:800

by [email protected] » Thu Nov 26, 2015 10:53 am
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
Contact Rich at [email protected]
Image
Top

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 16207
Joined: Mon Dec 08, 2008 6:26 pm
Location: Vancouver, BC
Thanked: 5254 times
Followed by:1268 members
GMAT Score:770

by Brent@GMATPrepNow » Thu Nov 26, 2015 11:06 am
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
Brent Hanneson - Creator of GMATPrepNow.com
Image
Top
Senior | Next Rank: 100 Posts
Posts: 42
Joined: Fri Oct 30, 2015 9:01 am

by shahfahad » Thu Nov 26, 2015 11:21 am
So we have to write the equation in terms of the constant:

for example the equation would be: p + k = r, r + k = s. But you have written in the TERMS of constant because when you will test the statements below, you can check that whether the sequence continues in TERMS of the constant. The constant value added to each of the equation in the sequence should be the same?

For example "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. "

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

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 16207
Joined: Mon Dec 08, 2008 6:26 pm
Location: Vancouver, BC
Thanked: 5254 times
Followed by:1268 members
GMAT Score:770

by Brent@GMATPrepNow » Thu Nov 26, 2015 11:29 am
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
Brent Hanneson - Creator of GMATPrepNow.com
Image
Top
GMAT Instructor
Posts: 2630
Joined: Wed Sep 12, 2012 3:32 pm
Location: East Bay all the way
Thanked: 625 times
Followed by:119 members
GMAT Score:780

by Matt@VeritasPrep » Fri Nov 27, 2015 12:02 am
shahfahad wrote:So we have to write the equation in terms of the constant:

for example the equation would be: p + k = r, r + k = s. But you have written in the TERMS of constant because when you will test the statements below, you can check that whether the sequence continues in TERMS of the constant. The constant value added to each of the equation in the sequence should be the same?

For example "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. "

Each equation equals to 2k. If one of it equals 2k and the next sequence equals k. Then it would not be arithmetic?
It might be easier to think of an arithmetic sequence as a sequence in which you generate one term by adding some constant to the term that came before it.

For instance,

1, 3, 5, 7, 9, 11, ...

is an arithmetic sequence, since I'm adding 2 to each term to form the next one.

A sequence like

1, 4, 9, 16, ...

is NOT an arithmetic sequence, however, since I'm adding DIFFERENT values each time (to 1 I add 3, but to 4 I add 5).
Top
Post new topic Post Reply
• Page 1 of 1