A certain infinite geometric series has first term 7 and sum 4. What is the result when the third term is divided by the second term?
Remember that r is the ratio between successive terms.
As for the infinite sum, remember that
S = a/(1-r), so
7/(1-r) = 4
To find the result when the third term is divided by the second term in an infinite geometric series, we first need to determine the common ratio (r).
We are given the first term (a₁) as 7 and the sum of the series (S) as 4. In an infinite geometric series, the sum converges to a value when the absolute value of the common ratio (|r|) is less than 1.
The formula to calculate the sum of an infinite geometric series is:
S = a₁ / (1 - r)
Now we can solve for r:
4 = 7 / (1 - r)
To isolate r, multiply both sides of the equation by (1 - r):
4(1 - r) = 7
Expand the equation:
4 - 4r = 7
Rearrange the terms:
-4r = 7 - 4
-4r = 3
Now divide both sides by -4 to solve for r:
r = -3/4
Now that we have the common ratio (r), we can find the third term (a₃) by multiplying the first term (a₁) by the common ratio squared (r²):
a₃ = a₁ * r²
a₃ = 7 * (-3/4)²
a₃ = 7 * 9/16
a₃ = 63/16
Finally, to find the result when the third term is divided by the second term, we divide a₃ by a₂:
a₃/a₂ = (63/16) / 7
a₃/a₂ = 63/112
Therefore, the result when the third term is divided by the second term in the given infinite geometric series is 63/112.