OG12: An arithmetic sequence

[nhai2003]

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 following must also be an arithmetic sequence?
p,r,s,t,u

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 help me with this one, thanhs guys!

[vivekjaiswal]

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 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.p2,r2,s2,t2,u2

A. I only
B.II only
C.III only
D. I and II
E. II and III
Please help me with this one, thanhs guys!

Hi nhai2003,

It seems you have missed out the list of letters as mentioned in the question. Also I believe the last term is III. p^2, r^2...

Cheers,
Vivek[/b]

[nhai2003]

thanks, now I understand the answer!

[Jeff@TargetTestPrep]

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 following must also be an arithmetic sequence?
p,r,s,t,u

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

Let's let the sequence be:

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

We can now use these numbers in the sequences to ascertain the truth or falseness of each statement. .

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 common 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 common 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 common difference between each pair of successive terms in the set. Thus, statement III is NOT true.

Answer: D

[Brent@GMATPrepNow]

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 following must also be an arithmetic sequence?
p,r,s,t,u

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 help me with this one, thanhs guys!

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 sequences

Answer: D

Cheers,
Brent