Question 4
a. Show that the following identity always holds:
$$a^{n+1}+b^{n+1}=(a+b)(a^n+b^n)-ab(a^{n-1}+b^{n-1}).$$
b. Let $a,b$ be real numbers with $a+b=1$ and $ab=-1$. Show that $a^{10}+b^{10}$ is an integer and calculate it.
$$a^{n+1}+b^{n+1}=(a+b)(a^n+b^n)-ab(a^{n-1}+b^{n-1}).$$
b. Let $a,b$ be real numbers with $a+b=1$ and $ab=-1$. Show that $a^{10}+b^{10}$ is an integer and calculate it.