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Arithmetic sequences S1 and S2 have 5 terms each. If the difference be

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Arithmetic sequences S1 and S2 have 5 terms each. If the difference be [#permalink]

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New post 26 Jan 2018, 11:40
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Arithmetic sequences S1 and S2 have 5 terms each. If the difference between two consecutive terms of S2 is twice the difference between two consecutive terms of S1, what is the ratio of the fifth term of S1 and the fifth term of S2?

(1) The first term of S1 is twice the first term of S2

(2) The second term of both S1 and S2 is 3
[Reveal] Spoiler: OA

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Arithmetic sequences S1 and S2 have 5 terms each. If the difference be [#permalink]

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New post 26 Jan 2018, 16:41
SandhyAvinash wrote:
Arithmetic sequences S1 and S2 have 5 terms each. If the difference between two consecutive terms of S2 is twice the difference between two consecutive terms of S1, what is the ratio of the fifth term of S1 and the fifth term of S2?

(1) The first term of S1 is twice the first term of S2

(2) The second term of both S1 and S2 is 3

Dear SandhyAvinash,

I'm happy to respond. :-)

To tell you the truth, I am little suspicious of this question, because I don't believe the GMAT expect students to recognize the term "arithmetic sequence" and understand its implications. An arithmetic sequence is one in which each term differs from the previous term by the addition of a fixed difference. If S1 has a fixed difference of d, then the prompt tells us that S2 has a fixed difference of 2d--these are the amounts added to get from one term to the next.

The starting values of the two sequences are unknown. The prompt tells us that the two sequences are of the form.

S1 = {a, a+d, a+2d, a+3d, a+4d}
S2 = {b, b+2d, b+4d, b+6d, b+8d}

Notice that there are three unknowns.

(1) The first term of S1 is twice the first term of S2

We are told a = 2b. This gives us one equation for three variables. We can't solve. The first statement, alone and by itself, is insufficient.

(2) The second term of both S1 and S2 is 3

a+d = 3
b + 2d = 3

This statement alone gives us two equation for three unknowns. We still can't solve. The second statement, alone and by itself, is insufficient.

Combined:
Now we have three equations:
a = 2b
a + d = 3
b + 2d = 3
Three equations for three unknowns. We can solve for everything. Combined, the statements are sufficient.

OA = (C)

It's not needed for the solution to the DS problem, but here's the solution:
Equation #1: a = 2b
Equation #2: a + d = 3
Equation #3: b + 2d = 3

Substitute equation #1 into equation #2 to eliminate a and get two equations for b and d.
2b + d = 3
Multiply this by two and subtract equation #3
(4b + 2d = 6)
-(b + 2d = 3)
3d = 3
d = 1

Substituting into equation #3, we get b = 1.
Substituting into equation #1 or #2, we get a = 2.

S1 = {2, 3, 4, 5, 6}
S2 = {1, 2, 5, 6, 9)


ratio requested = \(\tfrac{6}{9}\) = \(\tfrac{2}{3}\)

Does all this make sense?
Mike :-)
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Magoosh Test Prep

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Re: Arithmetic sequences S1 and S2 have 5 terms each. If the difference be [#permalink]

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New post 26 Jan 2018, 19:43
mikemcgarry wrote:
SandhyAvinash wrote:
Arithmetic sequences S1 and S2 have 5 terms each. If the difference between two consecutive terms of S2 is twice the difference between two consecutive terms of S1, what is the ratio of the fifth term of S1 and the fifth term of S2?

(1) The first term of S1 is twice the first term of S2

(2) The second term of both S1 and S2 is 3

Dear SandhyAvinash,

I'm happy to respond. :-)

To tell you the truth, I am little suspicious of this question, because I don't believe the GMAT expect students to recognize the term "arithmetic sequence" and understand its implications. An arithmetic sequence is one in which each term differs from the previous term by the addition of a fixed difference. If S1 has a fixed difference of d, then the prompt tells us that S2 has a fixed difference of 2d--these are the amounts added to get from one term to the next.

The starting values of the two sequences are unknown. The prompt tells us that the two sequences are of the form.

S1 = {a, a+d, a+2d, a+3d, a+4d}
S2 = {b, b+2d, b+4d, b+6d, b+8d}

Notice that there are three unknowns.

(1) The first term of S1 is twice the first term of S2

We are told a = 2b. This gives us one equation for three variables. We can't solve. The first statement, alone and by itself, is insufficient.

(2) The second term of both S1 and S2 is 3

a+d = 3
b + 2d = 3

This statement alone gives us two equation for three unknowns. We still can't solve. The second statement, alone and by itself, is insufficient.

Combined:
Now we have three equations:
a = 2b
a + d = 3
b + 2d = 3
Three equations for three unknowns. We can solve for everything. Combined, the statements are sufficient.

OA = (C)

It's not needed for the solution to the DS problem, but here's the solution:
Equation #1: a = 2b
Equation #2: a + d = 3
Equation #3: b + 2d = 3

Substitute equation #1 into equation #2 to eliminate a and get two equations for b and d.
2b + d = 3
Multiply this by two and subtract equation #3
(4b + 2d = 6)
-(b + 2d = 3)
3d = 3
d = 1

Substituting into equation #3, we get b = 1.
Substituting into equation #1 or #2, we get a = 2.

S1 = {2, 3, 4, 5, 6}
S2 = {1, 2, 5, 6, 9)


ratio requested = \(\tfrac{6}{9}\) = \(\tfrac{2}{3}\)

Does all this make sense?
Mike :-)


Thanks a lot, Mike, your way of explaining things is fantastic
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Kudos APPRECIATED!

Re: Arithmetic sequences S1 and S2 have 5 terms each. If the difference be   [#permalink] 26 Jan 2018, 19:43
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