If-a-1-2-a-2-3-a-n-2-a-n-1-a-2-find-a-n-

Question Number 128368 by I want to learn more last updated on 06/Jan/21

If a_{1} = 2, a_{2} = 3, a_{n + 2} = a_{n + 1} + \frac{a}{2}, find a_{n}

Answered by mr W last updated on 06/Jan/21

t h i s i s a n A P w i t h d = \frac{a}{2} = 3 - 2 = 1

\Rightarrow a_{n} = n + 1

Commented by I want to learn more last updated on 06/Jan/21

Thanks sir

Answered by mathmax by abdo last updated on 06/Jan/21

a_{n + 2} - a_{n + 1} = \frac{a}{2} \Rightarrow a_{n + 1} - a_{n} = \frac{a}{2} \Rightarrow \sum_{k = 1}^{n} (a_{n + 1} - a_{n}) = \frac{na}{2} \Rightarrow

a_{2} - a_{1} + a_{3} - a_{2} + \dots a_{n + 1} - a_{n} = \frac{na}{2} \Rightarrow

a_{n + 1} - a_{1} = \frac{na}{2} \Rightarrow a_{n + 1} = a_{1} + \frac{na}{2} \Rightarrow a_{n} = a_{1} + \frac{(n - 1) a}{2}

a_{2} = a_{1} + \frac{a}{2} \Rightarrow \frac{a}{2} = a_{2} - a_{1} = 1 \Rightarrow a = 2 \Rightarrow a_{n} = a_{1} + n - 1 = 2 + n - 1

\Rightarrow ★ a_{n} = n + 1 ★

Commented by I want to learn more last updated on 07/Jan/21

Thanks sir