Answer: the common ratio (r) is -2/5, |r| is less than 1, and the sum of the given series Σn=2^n/(-5)^n+1 is -4/175.
Step-by-step explanation:
In the given geometric series Σn=2^n/(-5)^(n+1), we can determine the common ratio (r) and the condition for convergence by analyzing the ratio of consecutive terms.
The general form of a geometric series is Σn=0 to ∞ ar^n, where a is the first term and r is the common ratio.
Comparing the given series to the general form, we have:
a = 2^2/(-5)^3 = 4/(-125) = -4/125
To find the common ratio (r), we divide the (n+1)th term by the nth term:
r = (2^(n+1))/(-5)^(n+2) divided by 2^n/(-5)^(n+1)
= (2^(n+1))*((-5)^(n+1))/((-5)^(n+2))*2^n
= 2/(-5)
= -2/5
To ensure convergence, we need the absolute value of the common ratio (|r|) to be less than 1.
|r| = |-2/5| = 2/5 < 1
Since |r| is less than 1, the given series Σn=2^n/(-5)^n+1 converges.
To determine the sum of the series, we use the formula for the sum of an infinite geometric series:
Sum = a/(1 - r)
Plugging in the values, we have:
Sum = (-4/125)/(1 - (-2/5))
= (-4/125)/(1 + 2/5)
= (-4/125)/(5/5 + 2/5)
= (-4/125)/(7/5)
= (-4/125) * (5/7)
= -20/875
= -4/175