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

Question
Series
asked 2020-12-09
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

Answers (1)

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.
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