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?
7/(1-r) = 4
7 = 4-4r
r = -3/4
The second part is now obvious, right?
To find the result when the third term is divided by the second term in the infinite geometric series, we need to determine the common ratio of the series.
Let's denote the first term of the series as a₁ = 7. The sum of an infinite geometric series can be calculated using the formula:
S = a / (1 - r)
Where S represents the sum of the series, a is the first term, and r is the common ratio.
In this case, we know that the sum of the series is 4. Therefore, we can substitute these values into the formula:
4 = 7 / (1 - r)
To solve for the common ratio, we need to rearrange the equation:
4(1 - r) = 7
4 - 4r = 7
-4r = 7 - 4
-4r = 3
Dividing both sides by -4 gives:
r = -3/4
Now that we know the common ratio is -3/4, we can calculate the third term of the series by multiplying the first term by the common ratio squared:
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:
a₃ / a₂ = (63/16) / 7
a₃ / a₂ = (63/16) * (1/7)
a₃ / a₂ = 9/32
Therefore, the result when the third term is divided by the second term is 9/32.