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25 Feb 2025, 22:57
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\(a_2\): 3
\(a_8\): 47
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Difficulty:
5%
(low)
Question Stats:
86% (01:54) correct
14%
(02:08)
wrong
based on 293
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For the sequence \(a_{1}, a_{2}, a_{3}\), ..., \(a_{n}, a_{n} = a_{n−1} + a_{n−2}\) for all values \(n>2\). If \(a_{1} = 1\) and \(a_{3} = 4\), what are the values of \(a_{2}\) and \(a_{8}\)?
| \(a_2\) | \(a_8\) | Answer values |
| 2 | ||
| 3 | ||
| 29 | ||
| 47 | ||
| 76 |
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Official Answer
\(a_2\): 3
\(a_8\): 47
poojaarora1818
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25 Feb 2025, 23:43
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Bismuth83
Solution:
As we know n>2, a1=1, a3=4
Here we go: a3 = a2 + a1 ---> 4 = a2 + 1 = 3
a4 = a3 + a2----> 4 + 3 = 7
a5 = a4 + a3----> 7 + 4 = 4
a6 = a5 + a4----->11+7 = 18
a7 = a6 + a5----->18+11= 29
a8 = a7 + a6----->29+18=47
So, a2 = 3 and a8 = 47
27 Feb 2025, 23:00
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1. We are asked to figure out the values of two specific terms in a given sequence.
2. The recursive identity has three consecutive terms, which means figuring out 3 neighboring terms will give us the whole sequence.
3. Since \(a_1\) and \(a_3\) are given, using our logic, it's best to find \(a_2\): \(a_3 = a_2 + a_1 \rightarrow 4 = a_2 + 1 \rightarrow a_2 = 3\).
4. Now we can find the following terms.
- \(a_4 = a_3 + a_2 = 4 + 3 = 7\).
- \(a_5 = a_4 + a_3 = 7 + 4 = 11\).
- \(a_6 = a_5 + a_4 = 11 + 7 = 18\).
- \(a_7 = a_6 + a_5 = 18 + 11 = 29\).
- \(a_8 = a_7 + a_6 = 29 + 18 = 47\).
5. Our answer will be: \(a_2 = 3\) and \(a_3 = 47\).
2. The recursive identity has three consecutive terms, which means figuring out 3 neighboring terms will give us the whole sequence.
3. Since \(a_1\) and \(a_3\) are given, using our logic, it's best to find \(a_2\): \(a_3 = a_2 + a_1 \rightarrow 4 = a_2 + 1 \rightarrow a_2 = 3\).
4. Now we can find the following terms.
- \(a_4 = a_3 + a_2 = 4 + 3 = 7\).
- \(a_5 = a_4 + a_3 = 7 + 4 = 11\).
- \(a_6 = a_5 + a_4 = 11 + 7 = 18\).
- \(a_7 = a_6 + a_5 = 18 + 11 = 29\).
- \(a_8 = a_7 + a_6 = 29 + 18 = 47\).
5. Our answer will be: \(a_2 = 3\) and \(a_3 = 47\).
