# What is the sum of the first six terms of the series? 40−10+2.5−0.625+... What is the answer as a simplified fraction

What is the sum of the first six terms of the series?
$40-10+2.5-0.625+...$
What is the answer as a simplified fraction
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odgovoreh
Given Data:
Series: $40-10+2.5-0.625+...$
The first term of series is: a=40
The second term of series is: ${a}_{2}=-10$
The third term of series is: ${a}_{3}=2.5$
For the first and second term,
The common ratio of the series is,
$r=\frac{{a}_{2}}{a}$
Substitute the values in the above equation.
$r=\frac{-10}{40}$
$=-0.25$
For the second and third term
The common ratio of the series is,
$r=\frac{{a}_{3}}{{a}_{2}}$
Substitute the values in the above equation.
$r=\frac{2.5}{-10}$
$=-0.25$
So, the given series is a geometric series.
The sum of the first six terms of the geometric series is,
${S}_{6}=a\frac{\left(1-{r}^{6}\right)}{\left(1-r\right)}$
Substitute the values in the above equation.
${S}_{6}=\left(40\right)\frac{\left(1-\left(-0.25{\right)}^{6}\right)}{\left(1-\left(-0.25\right)\right)}$
$=40\frac{\left(0.9998\right)}{1.25}$
$=31.99$
Thus, the sum of the first six terms of the given geometric series is 31.99.
Jeffrey Jordon