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?

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 \(p = 22+21+20...+3+2+1 = 253\).