For 7.21, we need to just follow the recursive definition for the sequence. We are given $a_0 = 1$. Then ever previous term (according to the recursive definition) is the sum of all the previous terms: $$\begin{aligned} a_1 &= a_0 = 1 \\ a_2 &= a_0 + a_1 = 1 + 1 = 2 \\ a_3 &= a_0 + a_1 + a_2 = 1 + 1 + 2 = 4 \\ a_4 &= \ldots \end{aligned}$$ From this, we start seeing the pattern that actually all the later terms all are powers of 2.