# How do you find the explicit formula for the following sequence 10, 12.5, 15, 17.5,...?

Mar 25, 2016

${n}^{t h}$ term is $\left(7.5 + 2.5 n\right)$ and
sum of the series up to ${n}^{t h}$ term is $\frac{n}{2} \left(17.5 + 2.5 n\right)$

#### Explanation:

The sequence $10 , 12.5 , 15 , 17.5 , \ldots$ is arithmetic sequence with first term $a = 10$ and common difference $d = 12.5 - 10 = 2 , 5$

In an arithmetic sequence $\left\{a , a + d , a + 2 d , a + 3 d , \ldots \ldots .\right\}$

${n}^{t h}$ term is given by

$a + \left(n - 1\right) d = 10 + \left(n - 1\right) 2.5 = 10 + 2.5 n - 2.5 = \left(7.5 + 2.5 n\right)$

Sum of the series up to ${n}^{t h}$ term is given by

n/2(2a+(n-1)d)=n/2(2*10+(n-1)*2.5 or

$\frac{n}{2} \left(20 + 2.5 n - 2.5\right)$ or $\frac{n}{2} \left(17.5 + 2.5 n\right)$