The first term of a geometric series is 4 and the sum of the first three terms of the series is 64. Find the sum of the first eight terms of the series.
Start with the Sum of n terms first.
sum= firstterm* ((1-r^n)/(1-r))
so first term=4, and n=3, solve for r.
Now that r is known, use the formula again to solve for sum of eight terms.
To find the sum of the first eight terms of the geometric series, we first need to find the common ratio (r) of the series.
Let's denote the first term as a₁ and the common ratio as r.
Given that the first term of the series (a₁) is 4, we can write the terms of the series as follows:
a₁ = 4
a₂ = 4 * r
a₃ = 4 * r * r = 4 * r²
The sum of the first three terms of the series (S₃) is given as 64, so we can write:
S₃ = a₁ + a₂ + a₃ = 4 + 4r + 4r² = 64
Now, we need to find the value of r that satisfies this equation.
Rearranging the equation, we have:
4r² + 4r + 4 - 64 = 0
4r² + 4r - 60 = 0
Dividing the equation by 4, we get:
r² + r - 15 = 0
Now, we can solve this quadratic equation by factoring or using the quadratic formula.
Factoring the equation, we have:
(r + 5)(r - 3) = 0
This gives us two possible solutions:
r = -5 or r = 3
Since the common ratio r must be positive in a geometric series, we discard the solution r = -5.
Therefore, the common ratio of the series is r = 3.
Now, we can compute the sum of the first eight terms of the series (S₈).
The formula for the sum of the first n terms of a geometric series is given by:
Sₙ = a₁ * (1 - rⁿ) / (1 - r)
For our series, a₁ = 4 and r = 3, and we want to find S₈.
Substituting these values into the formula, we have:
S₈ = 4 * (1 - 3⁸) / (1 - 3)
Simplifying the expression, we get:
S₈ = 4 * (1 - 6561) / (1 - 3)
S₈ = 4 * (-6560) / (-2)
Cancelling out the common factors, we have:
S₈ = 2 * 6560
S₈ = 13120
Therefore, the sum of the first eight terms of the geometric series is 13120.