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Chapter 11: Problem 18

Evaluate the finite series for the specified number of terms. $$ 4+12+36+\ldots ; n=6 $$

Short Answer

Expert verified
The sum of the first 6 terms of the series is 1460.

Step by step solution

01

Identify values for a, r, and n

Our series begins with an initial term (a) of 4. Each successive term is obtained by multiplying the previous one by the common ratio (r), which is 3. We need the sum of the first 6 terms (n). So, we have \(a = 4\), \(r = 3\), and \(n = 6\).
02

Plug values into sum formula

Let's insert \(a = 4\), \(r = 3\), and \(n = 6\) into the formula \(S_n = a \frac{{1-r^n}}{{1-r}}\). Substituting, we get \(S_6 = 4 \frac{{1-3^6}}{{1-3}}\).
03

Perform the calculations

After calculations are done, \(S_6 = 4 \frac{{1-729}}{{-2}} = 4 \cdot 365 = 1460\).

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