# The first three terms of an arithmetic sequence are given by x, (2x-5), 8.6 ?

## a) determine the first term and the common difference for the sequence b) determine the 20th term of the sequence c) determine the sum of the first 20 terms of the series

A: First term is $6.2$, common difference is $1.2$.
B: $29$.
C: $352$.

#### Explanation:

Question A:
Since this is an arithmetic sequence, there are two sides of the equation:
$8.6 - \left(2 x - 5\right) = \left(2 x - 5\right) - x$
Simplify.
$8.6 - 2 x + 5 = 2 x - 5 - x$
$13.6 - 2 x = x - 5$
$18.6 = 3 x$
$x = 6.2$
Hence, the first term is $6.2$, so the average of $6.2$ and $8.6$ is $7.4$.
This means that the common difference is $1.2$.

Question B:
Let's consider terms as $t$. The third term, for example, is $8.6$.
Let's consider the answers as $a$. One formula that can be derived is ${a}_{t} = 1.2 t + 5$.
Therefore, the 20th term would be:
${a}_{20} = 1.2 \left(20\right) + 5 = 24 + 5 = 29$.

Question C:
The easiest way to find a sum would be with this formula.
$\frac{{a}_{1} + {a}_{t}}{2} \cdot t$
There are 20 terms in the sequence.
$\frac{{a}_{1} + {a}_{20}}{2} \cdot 20$
Substituting, you would get this:
$\frac{6.2 + 29}{2} \cdot 20$
Solve.
$\frac{35.2}{2} \cdot 20 = 35.2 \cdot 10 = 352$