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A sequence with \(a_1 = 1\) is defined by  the recurrence relation \(​​a_{n+1} = 2^na_n\)  for all natural numbers n. If \(a_{23} = 2^p\), then what is p?

 Aug 26, 2018
 #1
avatar+356 
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For \(a23\)\(n = 22\). We can set up the equation \({2}^{22}\times a22 = 2p\). We can see that \(a22 = 2^{21}\times a21\). We can keep doing this until we get: \(2^{22+21+20...+3+2+1} \times 1 = 2^p\), so \(p = 22+21+20...+3+2+1 = 253\)

 

- Daisy

 Aug 26, 2018
 #2
avatar+902 
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Thanks!

Lightning  Aug 26, 2018

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