Final answer:
To find the sum of the first 6 terms of a geometric sequence with the given third and sixth term, we start by determining the common ratio and first term using the given terms, then apply the geometric series sum formula.
Step-by-step explanation:
To find the sum of the first 6 terms of a geometric series when the third term is 18 and the sixth term is 32, we need to use the formula for the nth term of a geometric sequence: an = a1 × rn-1 where an is the nth term, a1 is the first term, and r is the common ratio.
Given that the third term (a3) is 18 and the sixth term (a6) is 32, we can set up the following equations:
1. 18 = a1 × r2
2. 32 = a1 × r5
By dividing the second equation by the first, we eliminate a1 and can solve for r. Once we have both a1 and r, we can find the sum of the first 6 terms using the formula for the sum of a geometric series: Sn = a1(1 - rn) / (1 - r) where Sn is the sum of the first n terms.
The steps are as follows:
- Divide the second equation by the first to find r.
- Solve for a1 using one of the equations.
- Calculate the sum using the formula for the sum of a geometric series.