Find the Sum of the Series 10-15+22.5-33.75+50.625

This is a geometric sequence since there is a common ratio between each term. In this case, multiplying the previous term in the sequence by ${\displaystyle -\frac{3}{2}}$ gives the next term. In other words, ${\displaystyle a_n=a_1{r}^{n-1}}$.

Geometric Sequence: ${\displaystyle r=-\frac{3}{2}}$

This is the form of a geometric sequence.

Substitute in the values of ${\displaystyle a_1=10}$ and ${\displaystyle r=-\frac{3}{2}}$.

Combine ${\displaystyle 10}$ and ${\displaystyle \frac{{(-3)}^{n-1}}{2^{n-1}}}$.

This is the formula to find the sum of the first ${\displaystyle n}$ terms of the geometric sequence. To evaluate it, find the values of ${\displaystyle r}$ and ${\displaystyle a_1}$.

Replace the variables with the known values to find ${\displaystyle S_5}$.

Dividing two negative values results in a positive value.

Multiply the numerator by the reciprocal of the denominator.

Combine ${\displaystyle 5}$ and ${\displaystyle \frac{55}{8}}$.

Multiply ${\displaystyle 5}$ by ${\displaystyle 55}$.