Write the first five terms of the sequence whose n^ th term is a … - Vidyadip

Question

Write the first five terms of the sequence whose $n^{th}$ term is $a _ { n } = n . \frac { n ^ { 2 } + 5 } { 4 }$ .

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Answer

Given: $a_{n}=n \cdot \frac{n^{2}+5}{4}$
Putting n = 1,2,3,4 and 5, we get,
$a_{1}=1 \frac{1^{2}+5}{4}$$=1 . \frac{1+5}{4}=\frac{6}{4}=\frac{3}{2}$
$a_{2}=2 \cdot \frac{2^{2}+5}{4}$$=2 . \frac{4+5}{4}=\frac{18}{4}=\frac{9}{2}$
$a_{3}=3 . \frac{3^{2}+5}{4}$$=3 . \frac{9+5}{4}=3 \times \frac{14}{4}$$=\frac{42}{4}=\frac{21}{2}$
$a_{4}=4 \cdot \frac{4^{2}+5}{4}$$=4 . \frac{16+5}{4}=\frac{84}{4}$ = 21
$a_{5}=5 \cdot \frac{5^{2}+5}{4}$$=5 . \frac{25+5}{4}=5 \times \frac{30}{4}$$=\frac{150}{4}=\frac{75}{2}$
Therefore, the first five terms are $\frac{3}{2}, \frac{9}{2}, \frac{21}{2}$, 21 and $\frac{75}{2}$

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