If a_1 = 2, a_2 = 3, a_(n + 2) = a_(n + 1) + (a/2), find a


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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

	$a_{n + 2} - a_{n + 1} = \frac{a}{2} \Rightarrow a_{n + 1} - a_{n} = \frac{a}{2} \Rightarrow \sum_{k = 1}^{n} (a_{n + 1} - a_{n}) = \frac{na}{2} \Rightarrow$ $a_{2} - a_{1} + a_{3} - a_{2} + . . . a_{n + 1} - a_{n} = \frac{na}{2} \Rightarrow$ $a_{n + 1} - a_{1} = \frac{na}{2} \Rightarrow a_{n + 1} = a_{1} + \frac{na}{2} \Rightarrow a_{n} = a_{1} + \frac{(n - 1) a}{2}$ $a_{2} = a_{1} + \frac{a}{2} \Rightarrow \frac{a}{2} = a_{2} - a_{1} = 1 \Rightarrow a = 2 \Rightarrow a_{n} = a_{1} + n - 1 = 2 + n - 1$ $\Rightarrow ★ a_{n} = n + 1 ★$
	Commented by I want to learn more last updated on 07/Jan/21

	$Thanks sir$
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