Substituting n= 1, 2, 3, 4, 5 we obtain nbsp; nbsp; a _ 1 =1. 1 ^ 2 +5 / 4 = 6 / 4 = 3 / 2 a _ 2 =2. 2 ^ 2 +5 / 4 =2. 9 / 4 = 9 / 2 ...

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Proof by mathematical induction

a n =n n 2 +5...

Question

$a_{n} = n \frac{n^{2} + 5}{4}$

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Solution

Substituting $n = 1, 2, 3, 4, 5$ we obtain
$a_{1} = 1. \frac{1^{2} + 5}{4} = \frac{6}{4} = \frac{3}{2} a_{2} = 2. \frac{2^{2} + 5}{4} = 2. \frac{9}{4} = \frac{9}{2} a_{3} = 3. \frac{3^{2} + 5}{4} = 3. \frac{14}{4} = \frac{21}{2} a_{4} = 4. \frac{4^{2} + 5}{4} = 21 a_{5} = 5. \frac{5^{2} + 5}{4} = 5. \frac{30}{4} = \frac{75}{2}$

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