Sure, I'd be happy to walk you through how to solve this problem.
This particular sequence of numbers is known as a geometric series. A geometric series is a series of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In this case, it is easy to see that the common ratio is 1/4 because each term is a quarter of the previous one.
The first term (denoted by 'a') is 144, the number of terms (denoted by 'n') is 8, and the common ratio (denoted by 'r') is 1/4.
The sum ('S') of the first n terms of a geometric series can be found using the formula:
S_n = a * (1 - r^n) / (1 - r)
Let's plug in the values.
S_8 = 144 * (1 - (1/4)^8) / (1 - 1/4)
Perform the exponent first:
(1/4)^8 = 0.0000152587890625
Now substitute this back into the equation:
S_8 = 144 * (1 - 0.0000152587890625) / (1 - 1/4)
Subtract within the parentheses:
144 * (0.9999847412109375) / (0.75)
Multiply within the parentheses on top:
S_8 = 143.99704229736328 / 0.75
Now divide:
S_8 = 191.99605639648438
We round this to the nearest integer because the problem asks for the nearest integer.
S_8 = 192
So, the sum of the first 8 terms of the given series, rounded to the nearest integer, is .
Answer:192