nhai2003 wrote: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)
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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) = 2
k
Likewise, 2s - 2r = 2(
s - r) = 2
k
And 2t - 2s = 2(
t - s) = 2
k
And so on.
Since each term is derived by adding 2
k 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
