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16 Sep 2014, 00:21
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The sequence \(a_1\), \(a_2\), ..., \(a_n\), ... is such that \(a_n = a_1 + (n - 1)d\) for all integers \(n > 1\) and a certain constant \(d\). If the sum of the second and fifth terms of the sequence is 8, and the sum of the third and seventh terms is 14, what is the value of the first term?
A. 3
B. 2
C. 1
D. -1
E. -3
A. 3
B. 2
C. 1
D. -1
E. -3
16 Sep 2014, 00:21
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Official Solution:
The sequence \(a_1\), \(a_2\), ..., \(a_n\), ... is such that \(a_n = a_1 + (n - 1)d\) for all integers \(n > 1\) and a certain constant \(d\). If the sum of the second and fifth terms of the sequence is 8, and the sum of the third and seventh terms is 14, what is the value of the first term?
A. 3
B. 2
C. 1
D. -1
E. -3
Given:
\(a_2+a_5=(a_1+d)+(a_1+4d)=2a_1+5d=8\);
\(a_3+a_7=(a_1+2d)+(a_1+6d)=2a_1+8d=14\);
Subtracting the first equation from the second, we get \(3d = 6\), which implies \(d = 2\). Substituting this into the first equation, we get \(2a_1 + 5*2 = 8\), resulting in \(a_1 = -1\).
Answer: D
The sequence \(a_1\), \(a_2\), ..., \(a_n\), ... is such that \(a_n = a_1 + (n - 1)d\) for all integers \(n > 1\) and a certain constant \(d\). If the sum of the second and fifth terms of the sequence is 8, and the sum of the third and seventh terms is 14, what is the value of the first term?
A. 3
B. 2
C. 1
D. -1
E. -3
Given:
\(a_2+a_5=(a_1+d)+(a_1+4d)=2a_1+5d=8\);
\(a_3+a_7=(a_1+2d)+(a_1+6d)=2a_1+8d=14\);
Subtracting the first equation from the second, we get \(3d = 6\), which implies \(d = 2\). Substituting this into the first equation, we get \(2a_1 + 5*2 = 8\), resulting in \(a_1 = -1\).
Answer: D
General Discussion
08 Oct 2023, 09:42
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
