Question

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a geometric sequence has a common ratio $r = 2$. the sum of the first 4 terms is 60. what is the first term of the sequence?

Explanation:

Step1: Recall sum of geometric series formula

The sum of the first $n$ terms of a geometric sequence is given by:
$$S_n = a_1 \frac{r^n - 1}{r - 1}$$
where $S_n$ is the sum of the first $n$ terms, $a_1$ is the first term, $r$ is the common ratio, and $r
eq 1$.

Step2: Substitute known values

We know $S_4 = 60$, $r = 2$, $n = 4$. Plug these into the formula:
$$60 = a_1 \frac{2^4 - 1}{2 - 1}$$

Step3: Simplify the expression

Calculate the numerator and denominator:
$$60 = a_1 \frac{16 - 1}{1}$$
$$60 = 15a_1$$

Step4: Solve for $a_1$

Divide both sides by 15:
$$a_1 = \frac{60}{15}$$

Answer:

4