Question 144
Let $\{a_n\}_{n \ge 1}$ be the sequence of positive real numbers satisfying $a_1 = 1$ and
$$
a_{n+1} = a_n + \sqrt{a_n + a_{n+1}}
$$
for $n \ge 1$. Let $b_n = a_{n+1} - a_n$ for $n \ge 1$.
a. Show that $b_n \ge 1$.
b. Show that $a_n = b_n(b_n - 1)/2$.
c. Express $a_n$ in terms of $n$.
d. Find the sum $S = a_1 + a_2 + \cdots + a_{60}$.
$$
a_{n+1} = a_n + \sqrt{a_n + a_{n+1}}
$$
for $n \ge 1$. Let $b_n = a_{n+1} - a_n$ for $n \ge 1$.
a. Show that $b_n \ge 1$.
b. Show that $a_n = b_n(b_n - 1)/2$.
c. Express $a_n$ in terms of $n$.
d. Find the sum $S = a_1 + a_2 + \cdots + a_{60}$.