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!
OG12: An arithmetic sequence
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Hi nhai2003,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?
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!
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]
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Let's let the sequence be: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
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
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Basically, an arithmetic sequence is a sequence in which each term can be calculated by adding some constant, k, to the preceding term.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!
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