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705-805 (Hard)|   Sequences|      
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In a sequence, S, the ratio of each successive term is fixed. If the 28 Nov 2021, 10:53
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46% (01:47) correct 54% (01:48) wrong based on 259 sessions

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In a sequence, S, the ratio of each successive term is fixed. If the first term is 5, then what is the second term of the sequence?

(1) The difference between the third term and the first term is 15.
(2) The difference between the fourth term and the first term is 35.
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In a sequence, S, the ratio of each successive term is fixed. If the 28 Jun 2026, 00:32
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In a sequence, S, the ratio of each successive term is fixed. If the first term is 5, then what is the second term of the sequence?

The sequence is:

5, 5r, 5r^2, 5r^3, ...

Question: What is 5r?

(1) The difference between the third term and the first term is 15.

5r^2 - 5 = 15
r^2 = 4
r = 2 or -2

So the second term could be 10 or -10.

Not sufficient.

(2) The difference between the fourth term and the first term is 35.

5r^3 - 5 = 35
r^3 = 8
r = 2

So the second term is 5r = 10.

Sufficient.

Answer: B.

shashank35
I think the question was quite ambigious , sequence yess but what sequence is it in AP or GP it was not mentioned so how we could even begin with or if it was intentional then E would have been the answer
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The wording is not ambiguous. “The ratio of each successive term is fixed” means that each term is obtained by multiplying the previous term by the same constant. That describes a geometric sequence, not an arithmetic sequence.

If it were an arithmetic sequence, the question would say that the difference between successive terms is fixed.
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In a sequence, S, the ratio of each successive term is fixed. If the 28 Nov 2021, 11:26
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Statement 1
first term = 5
third term = 5r^2
We need to find second term i.e 5*r

third term - first term = 15
5r^2 - 5 = 15
r^2 = 4
r = +- 2

sequence with r=+2
5,10,20

sequence with r=-2
5,-10,20

Therefore, we cannot be sure about the second term

Statement 2
first term = 5
fourth term = 5r^3
We need to find second term i.e 5*r

fourth term - first term = 35
r^3 - 1 = 7
r^3 =8
r = 2

The only possible value for second term = 10
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In a sequence, S, the ratio of each successive term is fixed. If the 20 Jun 2026, 05:52
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We would need to take modulus while taking the different right (since it is not mentioned whether it is an increasing or decreasing sequence). Hence in the 2nd scenario we would get r^3 = 8 or -6, both of which are possible and hence cannot concluded from statement B alone as well.
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Statement 1
first term = 5
third term = 5r^2
We need to find second term i.e 5*r

third term - first term = 15
5r^2 - 5 = 15
r^2 = 4
r = +- 2

sequence with r=+2
5,10,20

sequence with r=-2
5,-10,20

Therefore, we cannot be sure about the second term

Statement 2
first term = 5
fourth term = 5r^3
We need to find second term i.e 5*r

fourth term - first term = 35
r^3 - 1 = 7
r^3 =8
r = 2

The only possible value for second term = 10
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In a sequence, S, the ratio of each successive term is fixed. If the 26 Jun 2026, 09:20
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Hi dhwanilms,

You're showing exactly the right DS instinct: refusing to assume a sign and asking whether a second case sneaks in. Let me separate the two things that are getting tangled here.

Where the wording lands

The statement says the difference between the fourth term and the first term is 35 - stated in that order, as a positive value. The standard reading is fourth - first = 35, so:

- 5r3 - 5 = 35 → r3 = 8

There's no separate "-35" case being asserted; the problem is telling you the fourth term exceeds the first by 35.

The real reason Statement 2 is sufficient

Even if you set the wording aside, the deeper point is even power vs. odd power.

- Statement 1 gives r2 = 4. A square always has two roots: r = 2or r = -2. That's where the ambiguity comes from - the even power, not the word "difference." Both survive, giving t2 = 10 or -10. Not sufficient.
- Statement 2 gives r3 = 8. A cube has exactly one real root: r = 2. There is no r = -2, because (-2)3 = -8, not 8. So t2 = 10, one value only. Sufficient.

That's the asymmetry: a square root forks into ±, but a real cube root does not.

On your r3 = -6 case

That case only appears if you force 5r3 - 5 = -35, i.e. read the difference as possibly negative. But the statement fixes the difference as +35 in the order given, so that branch isn't part of the question. And notice it wouldn't help Statement 1 anyway - there the two answers come straight from the square root, with the difference sign playing no role.

Takeaway: even powers create the ± fork; a single real cube root doesn't. That's why Statement 2 pins t2 down and Statement 1 can't.

Answer: B

dhwanilms
We would need to take modulus while taking the different right (since it is not mentioned whether it is an increasing or decreasing sequence). Hence in the 2nd scenario we would get r^3 = 8 or -6, both of which are possible and hence cannot concluded from statement B alone as well.

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In a sequence, S, the ratio of each successive term is fixed. If the 27 Jun 2026, 20:40
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I think the question was quite ambigious , sequence yess but what sequence is it in AP or GP it was not mentioned so how we could even begin with or if it was intentional then E would have been the answer
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In a sequence, S, the ratio of each successive term is fixed. If the 07 Jul 2026, 08:36
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Sorry if this is a silly question, but why in case of sq root 4 we have taken 2 cases but only one possible case in cube root 8? I made the mistake of selected D as the answer as I did not consider the negative case of -2. Could you please clarify?
Thanks in advance!
Bunuel
In a sequence, S, the ratio of each successive term is fixed. If the first term is 5, then what is the second term of the sequence?

The sequence is:

5, 5r, 5r^2, 5r^3, ...

Question: What is 5r?

(1) The difference between the third term and the first term is 15.

5r^2 - 5 = 15
r^2 = 4
r = 2 or -2

So the second term could be 10 or -10.

Not sufficient.

(2) The difference between the fourth term and the first term is 35.

5r^3 - 5 = 35
r^3 = 8
r = 2

So the second term is 5r = 10.

Sufficient.

Answer: B.



The wording is not ambiguous. “The ratio of each successive term is fixed” means that each term is obtained by multiplying the previous term by the same constant. That describes a geometric sequence, not an arithmetic sequence.

If it were an arithmetic sequence, the question would say that the difference between successive terms is fixed.
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In a sequence, S, the ratio of each successive term is fixed. If the 07 Jul 2026, 08:53
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ZeroSquare
Sorry if this is a silly question, but why in case of sq root 4 we have taken 2 cases but only one possible case in cube root 8? I made the mistake of selected D as the answer as I did not consider the negative case of -2. Could you please clarify?
Thanks in advance!

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For r^2 = 4, both 2 and -2 work because:

2^2 = 4 and (-2)^2 = 4

But for r^3 = 8, only 2 works because:

2^3 = 8, but (-2)^3 = -8

So even powers can hide the sign, but odd powers keep the sign.
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