Can someone please show me step by step how to solve this, I've been stuck.
Find the common ration r of the geometric series an that satisfies the following conditions: c=4, S=16
I would reallyy appreciate it if someone could help me
generally the 1st term is called a, but if that's what c is, then
S = c/(1-r) so
4/(1-r) = 16
1-r = 1/4
r = 3/4
THnak you!
Sure! I can help you with that. To find the common ratio (r) of the geometric series, we need to use the given information: c = 4 (the first term) and S = 16 (the sum of the series).
Step 1: Recall the formula for the sum of a finite geometric series, which is S = a * (1 - r^n) / (1 - r), where S represents the sum, a is the first term, r is the common ratio, and n is the number of terms in the series.
Step 2: In this case, we are given that S = 16 and a = 4, so we can substitute these values into the formula: 16 = 4 * (1 - r^n) / (1 - r).
Step 3: We also know that c = a = 4, which means the first term of the series is 4. Therefore, the sum of the series can be written as S = c * (1 - r^n) / (1 - r).
Step 4: Now we need to solve for r. To do this, we can rearrange the formula to isolate r: 16 * (1 - r) = 4 * (1 - r^n).
Step 5: Distribute the terms on both sides: 16 - 16r = 4 - 4r^n.
Step 6: Combine like terms: 16r - 4r^n = 12.
Step 7: Factor out r: r (16 - 4r^(n-1)) = 12.
Step 8: Divide both sides by (16 - 4r^(n-1)): r = 12 / (16 - 4r^(n-1)).
Step 9: Substitute the given value of c = 4 into the equation: r = 12 / (16 - 4r^(n-1)).
Step 10: Plug in the value for S = 16: r = 12 / (16 - 4r^(n-1)) = 16.
At this point, we have reached an equation with r on both sides. To find the exact value of r, we would need additional information about the number of terms or the value of n. Without that information, we can't determine the exact value of r. However, you can still use this equation to solve for r numerically using approximation methods or a calculator.
I hope this explanation helps you understand how to solve the problem step by step.