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Updated on: 08 Dec 2023, 06:15
Originally posted by gracieguidotti on 30 Oct 2023, 17:20.
Last edited by Bunuel on 08 Dec 2023, 06:15, edited 3 times in total.
Last edited by Bunuel on 08 Dec 2023, 06:15, edited 3 times in total.
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E
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For each positive integer n, let \(a_n = n(-1)^n(x-1)\). If the sum of \(a_1\) through \(a_{20}\) is equal to 60, what is the value of x?
(A) -5
(B) -2
(C) 2
(D) 4
(E) 7
(A) -5
(B) -2
(C) 2
(D) 4
(E) 7
02 Jan 2024, 13:09
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samdoesthegmat
For each positive integer n, let \(a_n = n(-1)^n(x-1).\) If the sum of \(a_1\) through \(a_2_0\) is equal to 60, what is the value of x?
A. -5
B. -2
C. 2
D. 4
E. 7
A. -5
B. -2
C. 2
D. 4
E. 7
\(a_1 = 1(-1)^1(x-1) = -1(x-1)\)
\(a_2 = 2(-1)^2(x-1) = 2(x-1)\)
\(a_3 = 3(-1)^3(x-1) = -3(x-1)\)
\(a_4 = 4(-1)^4(x-1) = 4(x-1)\)
.
.
General Pattern
- \(a_n = -n(x-1)\) ⇒ n = odd
- \(a_n = n(x-1)\) ⇒ n = even
.
.
\(a_{19} = -19(x-1)\)
\(a_{20} = 20(x-1)\)
\(a_1 + a_2 + a_3 + ...+ a_{19}+a_{20} = -1(x-1) + 2 (x-1) -3 (x-1) + 4 (x-1) -5 ....... -19(x-1) + 20(x-1) = 60\)
\((x-1)(-1 + 2 -3 + 4 -5 + 6 ... -19 + 20) = 60\)
\((x-1)(1 + 1 + 1 + 1 + \text{upto 10 times}) = 60\)
\((x-1)(10) = 60\)
\((x-1) = 6\)
\(x = 7\)
Option E
General Discussion
05 Jan 2024, 05:09
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Bunuel
Please, where am I wrong ?
the sum of a1 through a20 is equal to 60 means the average of the set is 60/20 = 3
the average is also a10 + a11 = 3*2=6
Then 10(x-1)-11(x-1)=6 --> x=-5
Please, where am I wrong ?
the sum of a1 through a20 is equal to 60 means the average of the set is 60/20 = 3
the average is also a10 + a11 = 3*2=6
Then 10(x-1)-11(x-1)=6 --> x=-5
