determine the sum of the following geometric series. round your answer to 4 decimal places if necessary.
a. -1/32+1/16-...+256
b. 50 over sigma n=1 8(.5)^n-2
To determine the sum of a geometric series, we can use the formula:
Sum = a * (1 - r^n) / (1 - r)
Where:
- "Sum" represents the sum of the geometric series.
- "a" is the first term of the series.
- "r" is the common ratio between consecutive terms.
- "n" is the number of terms in the series.
Let's solve the questions using this formula:
a. -1/32 + 1/16 - ... + 256
Here, the first term (a) is -1/32, and the common ratio (r) is 1/2. We can see that the common ratio is multiplying the previous term by 1/2 to get the next term.
The last term is 256, but we need to determine the number of terms (n) to use in the formula. To find n, we can set up the equation:
256 = (-1/32) * (1/2)^n
To solve for n, we can solve this equation for n using logarithms. Taking the logarithm base 1/2 of both sides:
log(base 1/2) (256) = n
n = log(base 1/2) (256)
Using a calculator, we find that log(base 1/2) (256) = 8.
Now that we have the values of a, r, and n, we can substitute them into the formula:
Sum = (-1/32) * (1 - (1/2)^8) / (1 - 1/2)
Simplifying this expression, we get:
Sum = (-1/32) * (1 - 1/256) / (1 - 1/2)
Sum = (-1/32) * (255/256) / (1/2)
Calculating this expression, we find:
Sum = -255/64 ≈ -3.9844
Therefore, the sum of the geometric series is approximately -3.9844.
b. 50 / Σ (n = 1 to 8) (0.5)^(n-2)
Here, we are given a summation notation. We need to find the sum of the expression (0.5)^(n-2) for n ranging from 1 to 8.
Let's expand the expression for each term:
For n = 1: (0.5)^(1-2) = (0.5)^(-1) = 2
For n = 2: (0.5)^(2-2) = (0.5)^0 = 1
For n = 3: (0.5)^(3-2) = (0.5)^1 = 0.5
For n = 4: (0.5)^(4-2) = (0.5)^2 = 0.25
For n = 5: (0.5)^(5-2) = (0.5)^3 = 0.125
For n = 6: (0.5)^(6-2) = (0.5)^4 = 0.0625
For n = 7: (0.5)^(7-2) = (0.5)^5 = 0.03125
For n = 8: (0.5)^(8-2) = (0.5)^6 = 0.015625
Now, we can calculate the sum by adding up all these terms:
Sum = 50 * (2 + 1 + 0.5 + 0.25 + 0.125 + 0.0625 + 0.03125 + 0.015625)
Using a calculator, we find:
Sum = 50 * 3.984375 = 199.21875
Rounding to 4 decimal places, the sum of the geometric series is approximately 199.2188.