help me how to define themgive a recursive definaition of the sequence {$\displaystyle {a_n}$} , n=1,2,3,...
a) $\displaystyle a_n = 1+(-1)^n$
b) $\displaystyle a_n = n(n+1)$
c) $\displaystyle a_n = n^2$
(a)
The sequence alternates between $\displaystyle 0$ and $\displaystyle 2$ starting with $\displaystyle 0.$ The sum of two consecutive terms is $\displaystyle 2.$ Hence
$\displaystyle a_1=0,\quad a_{n+1}=2-a_n$
(b)
$\displaystyle a_1=2,\quad a_{n+1}=(n+1)(n+2)=n(n+1)\frac{n+2}n=a_n\frac{n+2} n$
(c)
I leave this one to you.