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# If r is a constant a_n = rn for all positive integers n, for how many

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Intern
Joined: 14 Sep 2017
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If r is a constant a_n = rn for all positive integers n, for how many  [#permalink]

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27 Nov 2017, 03:27
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If r is a constant and $$a_n = rn$$ for all positive integers n, for how many values of n is $$a_n < 100$$?

(1) $$a_{50} = 500$$

(2) $$a_{100} + a_{105} = 2050$$
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Joined: 02 Sep 2009
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If r is a constant a_n = rn for all positive integers n, for how many  [#permalink]

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27 Nov 2017, 03:48
If r is a constant and $$a_n = rn$$ for all positive integers n, for how many values of n is $$a_n < 100$$?

(1) $$a_{50} = 500$$:

Since $$a_n = rn$$, then $$a_{50} =r*50= 500$$. Hence, r = 10. We know everything about the sequence, so we can answer the question. Sufficient.

(2) $$a_{100} + a_{105} = 2050$$:

$$100r + 105r = 2050$$ --> r = 10. The same here. Sufficient.

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Re: If r is a constant a_n = rn for all positive integers n, for how many  [#permalink]

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02 Oct 2019, 01:24
Hi Bunuel,

Thanks for the explanation.

But how did you come up with

a100+a105=2050 => 100r+105r=2050?
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Re: If r is a constant a_n = rn for all positive integers n, for how many  [#permalink]

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02 Oct 2019, 01:41
1
DealMakerOne wrote:
Hi Bunuel,

Thanks for the explanation.

But how did you come up with

a100+a105=2050 => 100r+105r=2050?

Since $$a_n = rn$$, then $$a_{100} =100r$$ and $$a_{105} =105r$$
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Re: If r is a constant a_n = rn for all positive integers n, for how many  [#permalink]

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03 Oct 2019, 05:59
Bunuel wrote:
DealMakerOne wrote:
Hi Bunuel,

Thanks for the explanation.

But how did you come up with

a100+a105=2050 => 100r+105r=2050?

Since $$a_n = rn$$, then $$a_{100} =100r$$ and $$a_{105} =105r$$

Thanks so much. It's clear now.

Just simply replace a100 with 100r and a105 with 105r => 100r+105r=2050 => 205r=2050 => r=10.
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Re: If r is a constant a_n = rn for all positive integers n, for how many  [#permalink]

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08 Oct 2019, 23:28
Bunuel

Hi, I just have one confusion here.

The question has not specified anything regarding the construction of the equation. How did you assume that a50 = r*50?

As we can also assume r = 5 and a50 = r*25. Can you please help me understand the same.
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Re: If r is a constant a_n = rn for all positive integers n, for how many  [#permalink]

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08 Oct 2019, 23:30
nehasomani33 wrote:
Bunuel

Hi, I just have one confusion here.

The question has not specified anything regarding the construction of the equation. How did you assume that a50 = r*50?

As we can also assume r = 5 and a50 = r*25. Can you please help me understand the same.

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Re: If r is a constant a_n = rn for all positive integers n, for how many  [#permalink]

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08 Oct 2019, 23:33
Bunuel

My question was on this arrival - Since an=rnan=rn, then
Quote:
a50=r∗50
=500a50=r∗50=500
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Re: If r is a constant a_n = rn for all positive integers n, for how many  [#permalink]

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08 Oct 2019, 23:35
nehasomani33 wrote:
Bunuel

My question was on this arrival - Since an=rnan=rn, then
Quote:
a50=r∗50
=500a50=r∗50=500

Isn't it given in the stem?

If r is a constant and$$a_n = rn$$ for all positive integers n, for how many values of n is $$a_n < 100$$?
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Re: If r is a constant a_n = rn for all positive integers n, for how many  [#permalink]

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08 Oct 2019, 23:38
Bunuel

Sorry! I am still not clear. Cant r (a constant value) be anything such as 5 or 15 or (may be 10)
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Re: If r is a constant a_n = rn for all positive integers n, for how many  [#permalink]

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08 Oct 2019, 23:43
nehasomani33 wrote:
Bunuel

Sorry! I am still not clear. Cant r (a constant value) be anything such as 5 or 15 or (may be 10)

The stem says: $$a_n = r*n$$

If r is a constant and $$a_n = rn$$ for all positive integers n, for how many values of n is $$a_n < 100$$?

(1) says: $$a_{50} = 500$$. Since $$a_n = rn$$, then $$a_{50} =r*50= 500$$. $$r*50= 500$$. Hence, r = 10.

Can you please tell me which step there is unclear?
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Re: If r is a constant a_n = rn for all positive integers n, for how many   [#permalink] 08 Oct 2019, 23:43
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