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Manager  Joined: 06 Apr 2010
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The infinite sequence a1, a2, …, an, … is such that a1=2  [#permalink]

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Question Stats: 83% (02:14) correct 17% (02:22) wrong based on 807 sessions

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The infinite sequence $$a_1$$, $$a_2$$,…, $$a_n$$, … is such that $$a_1 = 2$$, $$a_2 = -3$$, $$a_3 = 5$$, $$a_4 = -1$$, and $$a_n = a_{n-4}$$ for n > 4. What is the sum of the first 97 terms of the sequence?

A. 72
B. 74
C. 75
D. 78
E. 80
Math Expert V
Joined: 02 Sep 2009
Posts: 60627
Re: The infinite sequence a1, a2, …, an, … is such that a1=2  [#permalink]

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udaymathapati wrote:
The infinite sequence a1, a2,…, an,… is such that a1 = 2, a2 = -3, a3 = 5, a4 = -1, and an = an-4 for n > 4. What is the sum of the first 97 terms of the sequence?
A. 72
B. 74
C. 75
D. 78
E. 80

Given: $$a_1=2$$, $$a_2=-3$$, $$a_3=5$$ and $$a_4=-1$$. Also $$a_n=a_{n-4}$$, for $$n>4$$.

Since $$a_n=a_{n-4}$$ then:
$$a_5=a_1=2$$;
$$a_6=a_2=-3$$;
$$a_7=a_3=5$$;
$$a_8=a_4=-1$$;
$$a_9=a_5=a_1=2$$;
and so on.

So we have groups of 4: {2, -3, 5, -1}, the sum of each of such group is $$2-3+5-1=3$$. 97 terms consist of 24 full groups plus $$a_{97}=a_1=2$$, so the sum of first 97 terms of the sequence is $$24*3+2=74$$.

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Re: The infinite sequence a1, a2, …, an, … is such that a1=2  [#permalink]

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the sequence is repeating after each 4 terms.

i.e 2,-3,5,-1 ,2,-3,5,-1 ,2,-3,5,-1 ,2,-3,5,-1 ....till 97th term.

Now (2,-3,5,-1 ),(2,-3,5,-1) ,(2,-3,5,-1 ),......................(2,-3,5,-1) ,1

97 = 24 *4 + 1

Means 24 similar like (2,-3,5,-1 ) and one more that is first term of the series

the sum of the first four term is = 3

Hence sum is 24*3 + 2 = 74

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##### General Discussion
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Re: The infinite sequence a1, a2, …, an, … is such that a1=2  [#permalink]

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ashiima wrote:
The infinite sequence a1, a2,�, an,� is such that a1 = 2, a2 = -3, a3 = 5, a4 = -1, and an =
an-4 for n > 4. What is the sum of the first 97 terms of the sequence?
A. 72
B. 74
C. 75
D. 78
E. 80

Given: a(n) = a(n-4) i.e. the nth term is same as (n-4)th term e.g. 5th term is same as 1st term. 6th term is same as 2nd term etc

The sequence becomes: 2, -3, 5, -1, 2, -3, 5, -1, 2, -3, 5, -1 ...
The sum of each group of 4 terms = 2-3+5-1 = 3
How many such groups of 4 are there in the first 97 terms? When you divide by 4, you get 24 as quotient and 1 as remainder.
This means you get 24 complete groups of 4 terms each and 1 extra term i.e. the 97th term.
Sum of 24 groups of 4 terms each = 24*3 = 72
The 97th term will be 2 since it is the first term of the next group of four terms.
Sum of first 97 terms = 72+2 = 74
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Re: The infinite sequence a1, a2, …, an, … is such that a1=2  [#permalink]

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The sum of each set of 4 is 3 so to find the answer find the floor of 97/4 (this is 24) this leaves one term unaccounted for so the equation becomes 24*3+2=74.
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Re: The infinite sequence a1, a2, …, an, … is such that a1=2  [#permalink]

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can you explain a bit more.
from a5 to a93 i get an AP for which the sum is = 4371
from a1 to a4, the sum = 3
Total sum= 4374.
what am i doing wrong?
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Re: The infinite sequence a1, a2, …, an, … is such that a1=2  [#permalink]

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Baten80 wrote:
The infinite sequence a1, a2,, an, is such that a1 = 2, a2 = -3, a3 = 5, a4 = -1, and a_n = a_n-4 for n > 4. What is the sum of the first 97 terms of the sequence?
A. 72
B. 74
C. 75
D. 78
E. 80

Given for n>4 => a_n = a_n-4 => a_5 = a_1 & a_9 = a_5 = a_1

so taking the above facts => (a_1+a_2+a_3+a_4+a_1+.....) => 24(a_1+a_2+a_3+a_4)+a_1 (since it is the repitition of the 4 terms so 24*4 = 96)

so adding the above we get => 74 (B)
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Re: The infinite sequence a1, a2, …, an, … is such that a1=2  [#permalink]

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1
Nice one, a pattern problem with an addition twist.

a1 = 2, a2 = -3, a3 = 5, a4 = -1
What an = a(n-4) means that from a5 onwards, every number in the sequence will be equal to 4 numbers preceding that number. Therefore, a5 = a1 = 2, a6 = a2 = -3,...

Notice that this is a 4-element set pattern. If we sum a1-a4, we get 3. What is the closet multiple of 4 to 97 - 96! So, there are 24 such complete sets. Added to that is 97th number, which is first in the sequence = 2. So, we get (24 x 3) + 2 = 74 -> B
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$$a_1 = 2$$

$$a_2 = -3$$

$$a_3 = 5$$

$$a_4 = -1$$

$$a_5 = a_{(5-4)} = a_1 = 2$$

$$a_6 = a_{(6-4)} = a_2 = -3$$

$$(a_1 + a_2 + a_3 + a_4) = (a_5 + a_6 + a_7 + a_8) = (a_9 + a_{10} .....)$$ and so on

Addition upto $$a_{96}$$ is repeated 24 times

$$(a_1 + a_2 + a_3 + a_4) = 2 - 3 + 5 - 1 = 3$$

$$a_1 + a_2 + a_3 + ........ + a_{96} = 3 * 24 = 72$$

$$(a_1+a_2+a_3+ ........ + a_{96}) + a_{97} = 72 + 2 = 74$$

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Re: The infinite sequence a1, a2, …, an, … is such that a1=2  [#permalink]

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From the equation, a5 = a1, a6 = a2, a7 = a3, a8 = a4 and again a9 =a5 =a1
So sum of 97 terms =sum of 96 terms +97th term
= 24*( a1 +a2 +a3 +a4) + a1
= 24*(2 -3 +5 -1) + 2
= 72 +2
= 74. Option B
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Re: The infinite sequence a1, a2, …, an, … is such that a1=2  [#permalink]

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2
udaymathapati wrote:
The infinite sequence $$a_1$$, $$a_2$$,…, $$a_n$$, … is such that $$a_1 = 2$$, $$a_2 = -3$$, $$a_3 = 5$$, $$a_4 = -1$$, and $$a_n = a_{n-4}$$ for n > 4. What is the sum of the first 97 terms of the sequence?

A. 72
B. 74
C. 75
D. 78
E. 80

We are given that a(1) = 2, a(2) = -3, a(3) = 5, and a(4) = -1.

Since a(n) = a(n-4), we see that:

a(5) = a(5-4) = a(1) = 2,

a(6) = a(6-4) = a(2) = -3,

a(7) = a(7-4) = a(3) = 5,

and a(8) = a(8-4) = a(4) = -1

As we can see, the terms repeat themselves in a cycle of 4.

Since the sum of the terms a(1) to a(4) inclusive is [2 + (-3) + 5 + (-1)] = 3, the sum of a(5) to a(8) will also be 3, as will the sum of a(9) to a(12), etc.

We see from each “grouping” of 4 numbers, we have a sum of 3, and from 1 to 96 inclusive there are:

(96 - 4)/4 + 1 = 92/4 + 1 = 24 such groupings. Thus, the sum of those 24 groupings is 24 x 3 = 72.

We must also add the value of a(97), which equals 2.

Thus, the sum of the first 97 terms is: 24 x 3 + 2 = 72 + 2 = 74.

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Re: The infinite sequence a1, a2, …, an, … is such that a1=2  [#permalink]

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udaymathapati wrote:
The infinite sequence $$a_1$$, $$a_2$$,…, $$a_n$$, … is such that $$a_1 = 2$$, $$a_2 = -3$$, $$a_3 = 5$$, $$a_4 = -1$$, and $$a_n = a_{n-4}$$ for n > 4. What is the sum of the first 97 terms of the sequence?

A. 72
B. 74
C. 75
D. 78
E. 80

"$$a_n = a_{n-4}$$ for n > 4." <--- what is the point of this information?
You could still solve for the problem without it, so what was the test makers intention on giving us it?

Anyway, Value of n = 1,2,3,4 | Value of An= (2) + (-3) + (5) + (-1) = 3; and for A5, A6, A7,A8, given n>4 = n-4, value for n again equals 1,2,3,4 | Value for An = 2) + (-3) + (5) + (-1) = 3
Therefore, the sum of the first 8 digits of 97 equals 6
97/8 = 12; 12 x 6 = 72
since the next digit in line is the start of a new cycle (i.e. equal to A1), that digits value is 2
thus 72 + 2 = 74 Answer (B)

But really, other than determining the next four digits, I do not know the purpose of the above piece of information; seeing as An-4 for the ninth digit would make A5, which no value was given, it was confusing at first - does the value of An, where n>=9, would repeat or not. I mean, giving the test structure, and cyclicality, I knew that I could stop at 8 and solve for it. However, it just seems weird since what is given really doesn't account for the value of A9-4 (A5) since they never give us the value of A5, you have to assume it repeats. It seems clearer to state this the question without An. But, I still solved it because I knew what they were looking for for this category of question.
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The infinite sequence a1, a2, …, an, … is such that a1=2  [#permalink]

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udaymathapati wrote:
The infinite sequence $$a_1$$, $$a_2$$,…, $$a_n$$, … is such that $$a_1 = 2$$, $$a_2 = -3$$, $$a_3 = 5$$, $$a_4 = -1$$, and $$a_n = a_{n-4}$$ for n > 4. What is the sum of the first 97 terms of the sequence?

A. 72
B. 74
C. 75
D. 78
E. 80

Can anyone provide me with a quality information(website, youtube etc) on GMAT sequences. I cant seem to decode such sort of questions Math Expert V
Joined: 02 Sep 2009
Posts: 60627
Re: The infinite sequence a1, a2, …, an, … is such that a1=2  [#permalink]

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dave13 wrote:
udaymathapati wrote:
The infinite sequence $$a_1$$, $$a_2$$,…, $$a_n$$, … is such that $$a_1 = 2$$, $$a_2 = -3$$, $$a_3 = 5$$, $$a_4 = -1$$, and $$a_n = a_{n-4}$$ for n > 4. What is the sum of the first 97 terms of the sequence?

A. 72
B. 74
C. 75
D. 78
E. 80

Can anyone provide me with a quality information(website, youtube etc) on GMAT sequences. I cant seem to decode such sort of questions 12. Sequences

For ALL other subjects check Ultimate GMAT Quantitative Megathread

(I pointed you to this topic multiple times)

Hope it helps.
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Re: The infinite sequence a1, a2, …, an, … is such that a1=2  [#permalink]

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ScottTargetTestPrep wrote:
udaymathapati wrote:
The infinite sequence $$a_1$$, $$a_2$$,…, $$a_n$$, … is such that $$a_1 = 2$$, $$a_2 = -3$$, $$a_3 = 5$$, $$a_4 = -1$$, and $$a_n = a_{n-4}$$ for n > 4. What is the sum of the first 97 terms of the sequence?

A. 72
B. 74
C. 75
D. 78
E. 80

We are given that a(1) = 2, a(2) = -3, a(3) = 5, and a(4) = -1.

Since a(n) = a(n-4), we see that:

a(5) = a(5-4) = a(1) = 2,

a(6) = a(6-4) = a(2) = -3,

a(7) = a(7-4) = a(3) = 5,

and a(8) = a(8-4) = a(4) = -1

As we can see, the terms repeat themselves in a cycle of 4.

Since the sum of the terms a(1) to a(4) inclusive is [2 + (-3) + 5 + (-1)] = 3, the sum of a(5) to a(8) will also be 3, as will the sum of a(9) to a(12), etc.

We see from each “grouping” of 4 numbers, we have a sum of 3, and from 1 to 96 inclusive there are:

(96 - 4)/4 + 1 = 92/4 + 1 = 24 such groupings. Thus, the sum of those 24 groupings is 24 x 3 = 72.

We must also add the value of a(97), which equals 2.

Thus, the sum of the first 97 terms is: 24 x 3 + 2 = 72 + 2 = 74.

Hello Sir Why here are you deducting 4 from 96 in the numerator (96 - 4)/4 + 1 = 92/4 + 1 = 24

why didnt you write 96 /4 + 1 Another question why did you take initially take 96 to calculate the sum and than added value of 97 .. why not to take 97 intially i would appreaciate explanation thank you ! Intern  B
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Re: The infinite sequence a1, a2, …, an, … is such that a1=2  [#permalink]

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ok can someone pls tell me why you can't use Sn = n/2 (a + l) in this problem? We have the value of n, a and l Re: The infinite sequence a1, a2, …, an, … is such that a1=2   [#permalink] 09 Jul 2019, 08:37
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