# 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

Question
Series
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

2020-12-10
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,
$$S_6=a\frac{(1-r^6)}{(1-r)}$$
Substitute the values in the above equation.
$$S_6=(40)\frac{(1-(-0.25)^6)}{(1-(-0.25))}$$
$$=40\frac{(0.9998)}{1.25}$$
$$=31.99$$
Thus, the sum of the first six terms of the given geometric series is 31.99.

### Relevant Questions

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