Final answer:
To determine the number of terms (n) in a geometric sequence that sums up to 163,835, with a first term of 5 and common ratio of 2, one must apply the sum formula for geometric series and solve for n, which yields an answer of n = 15.
Step-by-step explanation:
To find the value of n for the geometric series with a first term of 5 and a common ratio of 2 that sums to 163,835, we can use the formula for the sum of the first n terms of a geometric sequence:
Sₙ = a(1 - rn) / (1 - r), where Sn is the sum of the first n terms, a is the first term, and r is the common ratio.
Plugging our values into the formula we get:
163,835 = 5(1 - 2n) / (1 - 2).
Since the common ratio is 2, this simplifies to:
163,835 = 5(2n - 1).
Dividing both sides by 5, we have:
32,767 = 2n - 1.
Adding 1 to both sides:
32,768 = 2n, which is 2 to the power of 15.
Therefore, n equals 15.