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The general term for a sequence is given as
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13 May 2024, 23:56
13 May 2024, 23:56
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The general term for a sequence is given as \(a_n=a_{(n-1)} -a_{(n-2)}\). If \(a_1=-5\) and \(a_2=4\). What is the sum of the first 100 terms?
A. 10
B. 11
C. 12
D. 13
E. 14
A. 10
B. 11
C. 12
D. 13
E. 14
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The general term for a sequence is given as
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15 May 2024, 10:36
15 May 2024, 10:36
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Calculate some terms to identify a pattern.
\(a_3 = a_{(3-1)} - a_{(3-2)} = a_2 - a_1 = 4 - (-5) = 9 \)
\(a_4 = a_3 - a_2 = 9 - 4 = 5 \)
\(a_5 = a_4 - a_3 = 5 - 9 = -4 \)
\(a_6 = a_5 - a_4 = -4 - 5 = -9 \)
\(a_7 = a_6 - a_5 = -9 - (-4) = -5 \)
The first 6 terms are -5, 4, 9, 5, -4, -9 and its sum is 0. The pattern "resets" on the 7th term, i.e., equals the 1st term.
Thus, the sum of terms that are multiples of 6 is 0. I.e., the sum of the first 96 terms is also 0.
That leaves terms 97, 98, 99, and 100, which will be -5, 4, 9, and 5.
The sum of those 4 terms = -5 + 4 + 9 + 5 = 13
\(a_3 = a_{(3-1)} - a_{(3-2)} = a_2 - a_1 = 4 - (-5) = 9 \)
\(a_4 = a_3 - a_2 = 9 - 4 = 5 \)
\(a_5 = a_4 - a_3 = 5 - 9 = -4 \)
\(a_6 = a_5 - a_4 = -4 - 5 = -9 \)
\(a_7 = a_6 - a_5 = -9 - (-4) = -5 \)
The first 6 terms are -5, 4, 9, 5, -4, -9 and its sum is 0. The pattern "resets" on the 7th term, i.e., equals the 1st term.
Thus, the sum of terms that are multiples of 6 is 0. I.e., the sum of the first 96 terms is also 0.
That leaves terms 97, 98, 99, and 100, which will be -5, 4, 9, and 5.
The sum of those 4 terms = -5 + 4 + 9 + 5 = 13
Re: The general term for a sequence is given as
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19 May 2024, 05:16
19 May 2024, 05:16
1
nurirachel wrote:
Calculate some terms to identify a pattern.
\(a_3 = a_{(3-1)} - a_{(3-2)} = a_2 - a_1 = 4 - (-5) = 9 \)
\(a_4 = a_3 - a_2 = 9 - 4 = 5 \)
\(a_5 = a_4 - a_3 = 5 - 9 = -4 \)
\(a_6 = a_5 - a_4 = -4 - 5 = -9 \)
\(a_7 = a_6 - a_5 = -9 - (-4) = -5 \)
The first 6 terms are -5, 4, 9, 5, -4, -9 and its sum is 0. The pattern "resets" on the 7th term, i.e., equals the 1st term.
Thus, the sum of terms that are multiples of 6 is 0. I.e., the sum of the first 96 terms is also 0.
That leaves terms 97, 98, 99, and 100, which will be -5, 4, 9, and 5.
The sum of those 4 terms = -5 + 4 + 9 + 5 = 13
\(a_3 = a_{(3-1)} - a_{(3-2)} = a_2 - a_1 = 4 - (-5) = 9 \)
\(a_4 = a_3 - a_2 = 9 - 4 = 5 \)
\(a_5 = a_4 - a_3 = 5 - 9 = -4 \)
\(a_6 = a_5 - a_4 = -4 - 5 = -9 \)
\(a_7 = a_6 - a_5 = -9 - (-4) = -5 \)
The first 6 terms are -5, 4, 9, 5, -4, -9 and its sum is 0. The pattern "resets" on the 7th term, i.e., equals the 1st term.
Thus, the sum of terms that are multiples of 6 is 0. I.e., the sum of the first 96 terms is also 0.
That leaves terms 97, 98, 99, and 100, which will be -5, 4, 9, and 5.
The sum of those 4 terms = -5 + 4 + 9 + 5 = 13
I would add that this is a recursive sequence. Sequences that follow patterns like that in the question stem should immediately make students think about iterative calculations to find a pattern.
While that's not in itself a formula, it's a great example of the GRE test makers giving students a clue to quickly find a solution.
