A1. What is the first term of the recursively defined sequence with the formula $a_{n}=2 a_{n-1}+1$, if the fifth term is 7?
(A) 2
(B) 15
(C) $-\frac{7}{2}$
(D) $-\frac{1}{2}$
(E) -5
A1. We write $a_{5}=2 a_{4}+1$ and calculate $a_{4}=\frac{a_{5}-1}{2}=\frac{7-1}{2}=3$. Step by step, we calculate all terms up to the first term $a_{3}=\frac{a_{4}-1}{2}=1, a_{2}=\frac{a_{3}-1}{2}=0, a_{1}=\frac{a_{2}-1}{2}=-\frac{1}{2}$.