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Arithmrtic Series

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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
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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
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Thanks!

Lightning  Aug 26, 2018