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Manager  Joined: 02 Dec 2012
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Difficulty:   5% (low)

Question Stats: 87% (01:38) correct 13% (01:49) wrong based on 1813 sessions

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In a certain sequence, the term $$x_n$$ is given by the formula $$x_n=2*x_{n-1}-\frac{1}{2}*x_{n-2}$$ for all $$n\geq{2}$$. If $$x_0=3$$ and $$x_1=2$$, what is the value of $$x_3$$?

(A) 2.5
(B) 3.125
(C) 4
(D) 5
(E) 6.75
Math Expert V
Joined: 02 Sep 2009
Posts: 59561
Re: In a certain sequence, the term xn is given by the formula  [#permalink]

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In a certain sequence, the term $$x_n$$ is given by the formula $$x_n=2*x_{n-1}-\frac{1}{2}*x_{n-2}$$ for all $$n\geq{2}$$. If $$x_0=3$$ and $$x_1=2$$, what is the value of $$x_3$$?

(A) 2.5
(B) 3.125
(C) 4
(D) 5
(E) 6.75

We have a formula to calculate the value of the terms in the sequence starting from $$x_2$$: $$x_n=2*x_{n-1}-\frac{1}{2}*x_{n-2}$$. Hence:

$$x_2=2*x_{1}-\frac{1}{2}*x_{0}=2*2-\frac{1}{2}*3=\frac{5}{2}$$;

$$x_3=2*x_{2}-\frac{1}{2}*x_{1}=2*\frac{5}{2}-\frac{1}{2}*2=4$$.

Hope it's clear.
##### General Discussion
Math Expert V
Joined: 02 Sep 2009
Posts: 59561
Re: In a certain sequence, the term xn is given by the formula  [#permalink]

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Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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Manager  Joined: 28 Feb 2012
Posts: 103
GPA: 3.9
WE: Marketing (Other)
Re: In a certain sequence, the term xn is given by the formula  [#permalink]

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In a certain sequence, the term $$x_n$$ is given by the formula $$x_n=2*x_{n-1}-\frac{1}{2}*x_{n-2}$$ for all $$n\geq{2}$$. If $$x_0=3$$ and $$x_1=2$$, what is the value of $$x_3$$?

(A) 2.5
(B) 3.125
(C) 4
(D) 5
(E) 6.75

X(3)= 2*X(2)-X(1)/2
X(2)=2X(1)-X(0)/2=2*2-3/2=5/2
Target Test Prep Representative V
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Re: In a certain sequence, the term xn is given by the formula  [#permalink]

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In a certain sequence, the term $$x_n$$ is given by the formula $$x_n=2*x_{n-1}-\frac{1}{2}*x_{n-2}$$ for all $$n\geq{2}$$. If $$x_0=3$$ and $$x_1=2$$, what is the value of $$x_3$$?

(A) 2.5
(B) 3.125
(C) 4
(D) 5
(E) 6.75

(Note that the * symbol in both the question stem and in this solution indicates multiplication.)

We are given that X(n) = 2 * X(n-1) – ½ * X(n-2), for all n=>2. This is called a recursive formula, which means that we need to know prior terms before we can compute the subsequent terms. For example, if we want to know X(2), we must know both X(1) and X(0), because X(2) is equal to 2 * X(1) – ½ * X(0).

We are given X(1) = 2 and X(0) = 3. So, when n is 2, X(2) would be calculated as follows:

X(2) = 2 * X(1) – ½ * X(0)

X(2) = 2 * 2 – ½ * 3

X(2) = 4 – 1.5

X(2) = 2.5

Now we are ready to determine the value of X(3). In this case, n = 3, X(1) is 2, and X(2) is 2.5. We plug these values into the recursive formula given in the question stem:

X(3) = 2 * X(2) – ½ * X(1)

X(3) = 2 * 2.5 – ½ * 2

X(3) = 5 – 1

X(3) = 4

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Intern  B
Joined: 29 Sep 2016
Posts: 7
Re: In a certain sequence, the term xn is given by the formula  [#permalink]

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If you have the OG 2017: No! you are not crazy it actually says the formula is used for all n>=2n. And on the answer choices it says n>= 2.

This made me take more than 10 minutes to solve because it just did not make sense. Cause if its n>= 2n that means that n = negative integer. BUT we don't have negative integers, so.... wthk am I supposed to do?

JUST WANTED TO LET YOU ALL KNOW! HUGEEEEE MISTAKE ON OG.
VP  D
Joined: 09 Mar 2016
Posts: 1229
In a certain sequence, the term xn is given by the formula  [#permalink]

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Bunuel wrote:
In a certain sequence, the term $$x_n$$ is given by the formula $$x_n=2*x_{n-1}-\frac{1}{2}*x_{n-2}$$ for all $$n\geq{2}$$. If $$x_0=3$$ and $$x_1=2$$, what is the value of $$x_3$$?

(A) 2.5
(B) 3.125
(C) 4
(D) 5
(E) 6.75

We have a formula to calculate the value of the terms in the sequence starting from $$x_2$$: $$x_n=2*x_{n-1}-\frac{1}{2}*x_{n-2}$$. Hence:

$$x_2=2*x_{1}-\frac{1}{2}*x_{0}=2*2-\frac{1}{2}*3=\frac{5}{2}$$;

$$x_3=2*x_{2}-\frac{1}{2}*x_{1}=2*\frac{5}{2}-\frac{1}{2}*2=4$$.

Hope it's clear.

Hello Bunuel, i always get confused with such long and weired formulas.

A few questions: How do you call mathematecally n-1 and n-2 written as subscript ? and another question how did you figure out that n-1 is 2 and n-2 is 3 when plugging in ? thank you Math Expert V
Joined: 02 Sep 2009
Posts: 59561
Re: In a certain sequence, the term xn is given by the formula  [#permalink]

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dave13 wrote:
Bunuel wrote:
In a certain sequence, the term $$x_n$$ is given by the formula $$x_n=2*x_{n-1}-\frac{1}{2}*x_{n-2}$$ for all $$n\geq{2}$$. If $$x_0=3$$ and $$x_1=2$$, what is the value of $$x_3$$?

(A) 2.5
(B) 3.125
(C) 4
(D) 5
(E) 6.75

We have a formula to calculate the value of the terms in the sequence starting from $$x_2$$: $$x_n=2*x_{n-1}-\frac{1}{2}*x_{n-2}$$. Hence:

$$x_2=2*x_{1}-\frac{1}{2}*x_{0}=2*2-\frac{1}{2}*3=\frac{5}{2}$$;

$$x_3=2*x_{2}-\frac{1}{2}*x_{1}=2*\frac{5}{2}-\frac{1}{2}*2=4$$.

Hope it's clear.

Hello Bunuel, i always get confused with such long and weired formulas.

A few questions: How do you call mathematecally n-1 and n-2 written as subscript ? and another question how did you figure out that n-1 is 2 and n-2 is 3 when plugging in ? thank you $$x_n=2*x_{n-1}-\frac{1}{2}*x_{n-2}$$ is a formula giving nth term in terms of (n-1)st and (n-2)nd terms. For example, it allows to find say, 3rd term if you know 3-1=2nd and 3-2=1st terms.
VP  D
Joined: 09 Mar 2016
Posts: 1229
Re: In a certain sequence, the term xn is given by the formula  [#permalink]

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Bunuel thanks! A few more questions if you dont mind What does for all n≥2 mean ?
Yes, you marked these in yellow x0=3 and x1=2
x0=3 this expression means that the zero term is 3 and x1=2 means that the first term is 2 correct?
If we need to find 3rd term isn’t it enough to know 2nd term ?
When you solve for x2, where from did you get 3 ?
Math Expert V
Joined: 02 Sep 2009
Posts: 59561
Re: In a certain sequence, the term xn is given by the formula  [#permalink]

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2
dave13 wrote:
Bunuel thanks! A few more questions if you dont mind What does for all n≥2 mean ?
Yes, you marked these in yellow x0=3 and x1=2
x0=3 this expression means that the zero term is 3 and x1=2 means that the first term is 2 correct?
If we need to find 3rd term isn’t it enough to know 2nd term ?
When you solve for x2, where from did you get 3 ?

1. It means that you can apply the given formula to calculate any term starting from x2;
2. Yes.
3. To get x3 you need x2 and x1 because the formula links nth term to TWO proceeding terms, (n-1)st and (n-2)nd

12. Sequences

Manager  B
Joined: 20 Apr 2019
Posts: 85
In a certain sequence, the term xn is given by the formula  [#permalink]

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The timer says 88% of people got it right but I bet I can count on my fingers the number who did it in two minutes.

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Non-Human User Joined: 09 Sep 2013
Posts: 13709
Re: In a certain sequence, the term xn  [#permalink]

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Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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_________________ Re: In a certain sequence, the term xn   [#permalink] 21 Aug 2019, 00:56
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