Chapter 8.2A, Problem 17A

In a Fibonacci-like sequence, the first term is $\sqrt{2}$, the second is $\sqrt{3}$, and each subsequent term is the sum of two previous terms. Find the following.

a. The first 10 terms in simplest form without computing

$\sqrt{2}$ or $\sqrt{3}$.

b. If the terms in the Fibonacci sequence are $F_1,F_2,F_3,\dots ,$ and the terms in the Fibonacci-like sequence in this question are $L_1,L_2,L_3,\dots ,$ answer the following.

i. Verify that $L_7=F_5\sqrt{2}+F_6\sqrt{3}$.

ii. Write $L_8,L_9\text{ and }{L}_{10}$ using the terms of the Fibonacci sequence, $\sqrt{2}$, and $\sqrt{3}$.

iii. Use your answers in the previous part to conjecture an expression for $L_n$ in terms of $\sqrt{2}$, $\sqrt{3}$, and Fibonacci terms.