# What is the sum of the first 4 terms of the arithmetic sequence in which the 6th term is 8 and the 10th term is 13?

May 5, 2016

Sum of the first 4 terms is $14.50$

#### Explanation:

In an arithmetic sequence, whose first term is $a$ and difference between a term and its preceding term is $d$,

the ${n}^{t h}$ term is $a + \left(n - 1\right) d$ and sum of first $n$ terms is $\frac{n}{2} \left(2 a + \left(n - 1\right) d\right)$

Hence ${6}^{t h}$ term will be $a + 5 d = 8$ and ${10}^{t h}$ term will be $a + 9 d = 13$

Subtracting first from second, $4 d = 5$ or $d = 1.25$

and $a = 8 - 5 \cdot 1.25 = 8 - 6.25 = 1.75$

Hence sum of first four terms is

$\frac{4}{2} \cdot \left(2 \cdot 1.75 + 3 \cdot 1.25\right) = 2 \cdot \left(3.5 + 3.75\right) = 2 \cdot 7.25 = 14.50$