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?
+26387 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 \(p\)?
\(\begin{array}{|rcll|} \hline a_1 &=& 1 \\\\ a_2 &=& 2^1+a_1 \\ &=& 2^1+1 \\ &=& 2^2-1 \\\\ a_3 &=& 2^2+a_2 \\ &=& 2^2+2^2-1 \\ &=& 2*2^2-1 \\ &=& 2^3-1 \\\\ a_4 &=& 2^3+a_3 \\ &=& 2^3+2^3-1 \\ &=& 2*2^3-1 \\ &=& 2^4-1 \\\\ a_5 &=& 2^4+a_4 \\ &=& 2^4+2^4-1 \\ &=& 2*2^4-1 \\ &=& 2^5-1 \\ && \dots \\ \mathbf{a_n} &=& \mathbf{2^n-1} \quad | \quad n = 23 \\\\ a_{23} &=& 2^{23} - 1 \\ \mathbf{a_{23}} &=& \mathbf{8388607} \\ \hline \end{array}\)
