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The sequence \(a_0, a_1, a_2, \ldots\,\) satisfies the recurrence equation \( a_n = 2 a_{n-1} - 2 a_{n - 2} + a_{n - 3}\) for every integer \(n \ge 3\). If \(a_{20} = 1, a_{25} = 10, \)and \(a_{30} = 100\), what is the value of \(a_{1331}\)?

 May 3, 2019
 #1
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a(25) - a(20) + 1 = 6 terms

The Ratio =(10/1)^1/6 =1.4677992676220695409205171148169

a(1331) - a(30) + 1 = 1,302 terms. Therefore the:

a(1331) =100 x 1.4677992676220695409205171148169^1302 =10^219 Exactly!.

 May 3, 2019
 #2
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Another way:

 

Since a(30) - a(25) + 1 = 6 terms which go up by a factor of 10, then:

a(1331) - a(30) + 1 =1,302 terms. Then we simply divide:

1,302 / 6 =217 6-term periods. And finally:

100 x 10^217 = 10^219 - which is a(1,331)'s term.

 May 4, 2019

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