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# In a certain sequence, the term an is defined by the formula a_n =

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Math Expert
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In a certain sequence, the term an is defined by the formula a_n =  [#permalink]

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30 Jul 2018, 00:57
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In a certain sequence, the term an is defined by the formula $$a_n = a_{n – 1} + 10$$ for each integer n ≥ 2. What is the positive difference between $$a_{10}$$ and $$a_{15}$$?

(A) 5
(B) 10
(C) 25
(D) 50
(E) 100

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30 Jul 2018, 04:16
Bunuel wrote:
In a certain sequence, the term an is defined by the formula $$a_n = a_{n – 1} + 10$$ for each integer n ≥ 2. What is the positive difference between $$a_{10}$$ and $$a_{15}$$?

An = An-1 + 10

A15 = A14 + 10
A15 = A13 + 10 + 10
A15 = A12 + 10 + 10 + 10
A15 = A11 + 10 + 10 + 10 + 10
A15 = A10 + 10 + 10 + 10 + 10 + 10
A15 - A10 = 10 + 10 + 10 + 10 + 10
A15 - A10 = 50

Hence, D.
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Re: In a certain sequence, the term an is defined by the formula a_n =  [#permalink]

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10 Aug 2018, 02:54
sudarshan22 wrote:
Bunuel wrote:
In a certain sequence, the term an is defined by the formula $$a_n = a_{n – 1} + 10$$ for each integer n ≥ 2. What is the positive difference between $$a_{10}$$ and $$a_{15}$$?

An = An-1 + 10

A15 = A14 + 10
A15 = A13 + 10 + 10
A15 = A12 + 10 + 10 + 10
A15 = A11 + 10 + 10 + 10 + 10
A15 = A10 + 10 + 10 + 10 + 10 + 10
A15 - A10 = 10 + 10 + 10 + 10 + 10
A15 - A10 = 50

Hence, D.

Thanks for the explanation, your approach absolutely makes sense to me.

Can somebody verify that the following approach is valid as well:

I know n is defined as 2 or bigger but plugging in 1 for n leaves as with a1 = a0 +10 and a0 has to be 0 as there is no 0th/st sequence.

Hence a1=10,a2=20,a3=30,a4=40,a5=50 and therefore a10 = 100 and a15 = 150

100-150 = -50 and since we are looking for the positive difference D is the final answer choice.

Max
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Joined: 18 Jun 2018
Posts: 267
Re: In a certain sequence, the term an is defined by the formula a_n =  [#permalink]

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10 Aug 2018, 03:02
1
maxmayr wrote:
sudarshan22 wrote:
Bunuel wrote:
In a certain sequence, the term an is defined by the formula $$a_n = a_{n – 1} + 10$$ for each integer n ≥ 2. What is the positive difference between $$a_{10}$$ and $$a_{15}$$?

An = An-1 + 10

A15 = A14 + 10
A15 = A13 + 10 + 10
A15 = A12 + 10 + 10 + 10
A15 = A11 + 10 + 10 + 10 + 10
A15 = A10 + 10 + 10 + 10 + 10 + 10
A15 - A10 = 10 + 10 + 10 + 10 + 10
A15 - A10 = 50

Hence, D.

Thanks for the explanation, your approach absolutely makes sense to me.

Can somebody verify that the following approach is valid as well:

I know n is defined as 2 or bigger but plugging in 1 for n leaves as with a1 = a0 +10 and a0 has to be 0 as there is no 0th/st sequence.

Hence a1=10,a2=20,a3=30,a4=40,a5=50 and therefore a10 = 100 and a15 = 150

100-150 = -50 and since we are looking for the positive difference D is the final answer choice.

Max

maxmayr

Quote:
In a certain sequence, the term an is defined by the formula $$a_n = a_{n – 1} + 10$$ for each integer n ≥ 2.

Though you are getting a correct answer,you are assuming $$a_n = a_{n – 1} + 10$$ is valid for n<2 also.
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Joined: 21 Jul 2017
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In a certain sequence, the term an is defined by the formula a_n =  [#permalink]

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10 Aug 2018, 04:05
Quote:
In a certain sequence, the term an is defined by the formula $$a_n = a_{n – 1} + 10$$ for each integer n ≥ 2.
Though you are getting a correct answer,you are assuming $$a_n = a_{n – 1} + 10$$ is valid for n<2 also.

Yes, absolutely true, but then I have to ask where do we get information to conclude that:

if A15 = A14 + 10, then A15 = A13 + 10 + 10

We substitute A14 automatically with 10.

What do I miss out here?

Kind regards,
Max
Senior Manager
Joined: 18 Jun 2018
Posts: 267
Re: In a certain sequence, the term an is defined by the formula a_n =  [#permalink]

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10 Aug 2018, 04:34
1
maxmayr wrote:
Quote:
In a certain sequence, the term an is defined by the formula $$a_n = a_{n – 1} + 10$$ for each integer n ≥ 2.
Though you are getting a correct answer,you are assuming $$a_n = a_{n – 1} + 10$$ is valid for n<2 also.

Yes, absolutely true, but then I have to ask where do we get information to conclude that:

if A15 = A14 + 10, then A15 = A13 + 10 + 10

We substitute A14 automatically with 10.

What do I miss out here?

Kind regards,
Max

$$A_n=A_{n-1}+10$$
$$A_{15}=A_{14} +10$$ ...(1)
$$A_{14}=A_{13} +10$$ ...(2)
$$A_{13}=A_{12} +10$$ ...(3)
$$A_{12}=A_{11} +10$$ ...(4)
$$A_{11}=A_{10} +10$$ ...(5)
Substituting (5) into (4)
We get
$$A_{12}= A_{10} +10+10$$ ...(6)
Substituting (6) into (3)
$$A_{13}=A_{10} +10+10 +10$$ ...(7)
Substituting (7) into (2)
$$A_{14}=A_{10} +10+10 +10 +10$$ ...(8)
Substituting (8) into (1)
$$A_{15}=A_{10} +10+10 +10 +10 +10$$
$$A_{15}=A_{10} +50$$
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Re: In a certain sequence, the term an is defined by the formula a_n =  [#permalink]

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10 Aug 2018, 04:38
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Re: In a certain sequence, the term an is defined by the formula a_n =   [#permalink] 10 Aug 2018, 04:38
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