Question

# Find the first six terms of the sequence defined by each of these recurrence relations and initial conditions. a) an = (an−1)2 , a1 = 2 ( Just find its general formula )

Sequences
Find the first six terms of the sequence defined by each of these recurrence relations and initial conditions.
a) an = (an−1)2 , a1 = 2 ( Just find its general formula )

2021-02-06
The given recurrence relation is: an=2an-1, a1=2
Note thet
a1=2
$$\displaystyle{a}{2}={2}{a}{1}={2}\cdot{2}={4}={2}^{{2}}$$
$$\displaystyle{a}{3}={2}{a}{2}={2}\cdot{4}={8}={2}^{{3}}$$
$$\displaystyle{a}{4}={2}{a}{3}={2}\cdot{8}={16}={2}^{{4}}$$
$$\displaystyle{a}{5}={2}{a}{4}={2}\cdot{16}={32}={2}^{{5}}$$
$$\displaystyle{a}{6}={2}{a}{5}={2}\cdot{32}={64}={2}^{{6}}$$
Therefore we find the general term is $$\displaystyle{a}{n}={2}^{{n}}$$, for $$\displaystyle{n}∈{N}$$. Prove: Note that
$$\displaystyle{a}{n}={2}{a}{n}-{1}={2}\cdot{2}\cdot{a}{n}-{2}={2}^{{2}}{a}{n}-{2}=\ldots={2}^{{n}}-{1}\cdot{a}{1}={2}^{{n}}$$