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# Sequence

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

Jun 9, 2021

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+26115
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A sequence with $$a_1 = 1$$ is defined by the recurrence relation
$$a_{n+1} = 2^n + a_n$$ for all natural numbers $$n$$.
If $$a_{23} = p$$, then what is $$p$$?

$$\begin{array}{|rcll|} \hline a_1 &=& 1 \\\\ a_2 &=& 2^1+a_1 \\ &=& 2^1+1 \\ &=& 2^2-1 \\\\ a_3 &=& 2^2+a_2 \\ &=& 2^2+2^2-1 \\ &=& 2*2^2-1 \\ &=& 2^3-1 \\\\ a_4 &=& 2^3+a_3 \\ &=& 2^3+2^3-1 \\ &=& 2*2^3-1 \\ &=& 2^4-1 \\\\ a_5 &=& 2^4+a_4 \\ &=& 2^4+2^4-1 \\ &=& 2*2^4-1 \\ &=& 2^5-1 \\ && \dots \\ \mathbf{a_n} &=& \mathbf{2^n-1} \quad | \quad n = 23 \\\\ a_{23} &=& 2^{23} - 1 \\ \mathbf{a_{23}} &=& \mathbf{8388607} \\ \hline \end{array}$$

Jun 9, 2021