To find the value of n, we need to first simplify the given equation:
sum k = 1 to n 3 * (2) ^ (1 - k) = 5.8125
=> sum k = 1 to n 3 * (2) / (2^k) = 5.8125
=> sum k = 1 to n 3 / (2^(k-1)) = 5.8125
=> 3/2^0 + 3/2^1 + 3/2^2 + ... + 3/2^(n-1) = 5.8125
Now, let's calculate the sum of the series on the left side:
3/2^0 + 3/2^1 + 3/2^2 + ... + 3/2^(n-1) = 3 * (1 + 1/2 + 1/4 + ... + 1/2^(n-1))
This is a geometric series with first term a = 1 and common ratio r = 1/2.
The sum of a geometric series up to n terms is given by:
S = a(1 - r^n) / (1 - r)
Plugging in the values of a = 1, r = 1/2, and S = 5.8125 into the formula, we get:
5.8125 = 1(1 - (1/2)^n) / (1 - 1/2)
5.8125 = (1 - (1/2)^n) / (1/2)
Solving for n:
11.625 = 1 - (1/2)^n
(1/2)^n = 1 - 11.625
(1/2)^n = -10.625
This equation does not have a real solution, meaning there is no value of n that satisfies the original equation.
c) Given: sum k = 1 to n 3 * (2) ^ (1 - k) = 5.8125
Calculate the value of n.
1 answer