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16 Sep 2014, 01:22
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The sequence \(a_1\), \(a_2\), \(a_3\), ... , \(a_n\), ... is such that \(a_n=a_{n-1}+a_{n-2}\) for all \(n \ge 3\). If \(a_1 = 5\) and \(a_5 = 7\), what is the value of \(a_2\) ?
A. -3
B. -1
C. 0
D. 2
E. 3
A. -3
B. -1
C. 0
D. 2
E. 3
16 Sep 2014, 01:22
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Official Solution:
The sequence \(a_1\), \(a_2\), \(a_3\), ... , \(a_n\), ... is such that \(a_n=a_{n-1}+a_{n-2}\) for all \(n \ge 3\). If \(a_1 = 5\) and \(a_5 = 7\), what is the value of \(a_2\) ?
A. -3
B. -1
C. 0
D. 2
E. 3
Since given that \(a_n=a_{n-1}+a_{n-2}\), then:
\(a_5=a_4+a_3=(a_3+a_2)+(a_2+a_1)=((a_2+a_1)+a_2)+(a_2+a_1)=3*a_2+2*a_1\).
So, we have that \(a_5=3*a_2+2*a_1\), or \(7=3*a_2+2*5\), which gives \(a_2=-1\).
Answer: B
The sequence \(a_1\), \(a_2\), \(a_3\), ... , \(a_n\), ... is such that \(a_n=a_{n-1}+a_{n-2}\) for all \(n \ge 3\). If \(a_1 = 5\) and \(a_5 = 7\), what is the value of \(a_2\) ?
A. -3
B. -1
C. 0
D. 2
E. 3
Since given that \(a_n=a_{n-1}+a_{n-2}\), then:
\(a_5=a_4+a_3=(a_3+a_2)+(a_2+a_1)=((a_2+a_1)+a_2)+(a_2+a_1)=3*a_2+2*a_1\).
So, we have that \(a_5=3*a_2+2*a_1\), or \(7=3*a_2+2*5\), which gives \(a_2=-1\).
Answer: B
General Discussion
11 Aug 2015, 04:08
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we can solve this by plugging answer options as well.
we can come up with this,
a1 = 5
a2
a3 = a2 + a1
a4 = a3 + a2
a5 = 7
Lets assume a2 = 2
a1 = 5
a2 = 2
a3 = 2 + 5 = 7
a4 = 7 + 2 = 9
a5 = 7 but a3 + a4 gives us 16...so a2 is not 2...
we can then try -1
Lets assume a2 = -1
a1 = 5
a2 = -1
a3 = -1 + 5 = 4
a4 = 4 + -1 = 3
a5 = 7
the sequence values are as expected. hence a2 = -1.
we can come up with this,
a1 = 5
a2
a3 = a2 + a1
a4 = a3 + a2
a5 = 7
Lets assume a2 = 2
a1 = 5
a2 = 2
a3 = 2 + 5 = 7
a4 = 7 + 2 = 9
a5 = 7 but a3 + a4 gives us 16...so a2 is not 2...
we can then try -1
Lets assume a2 = -1
a1 = 5
a2 = -1
a3 = -1 + 5 = 4
a4 = 4 + -1 = 3
a5 = 7
the sequence values are as expected. hence a2 = -1.
