In a geometric sequence, the ratio between consecutive terms remains constant. To find the 12th term in the sequence given that 84 is the 5th term and 27 is the 3rd term, we need to determine the common ratio first.
1. Calculate the common ratio (r) using the terms provided:
- For the term 84 (5th term), it means a = 84.
- For the term 27 (3rd term), it means a = 27.
- Using the formula for the nth term of a geometric sequence: an = a1 * r^(n-1), where an is the nth term, a1 is the first term, r is the common ratio, and n is the term number.
- From a5 = 84 and a3 = 27:
- 84 = a1 * r^(5-1) = a1 * r^4
- 27 = a1 * r^(3-1) = a1 * r^2
- Divide the equations to find the ratio r:
- 84 / 27 = (a1 * r^4) / (a1 * r^2)
- 3.1111... = r^2
- r ≈ √3.1111 ≈ 1.7632 (approx.)
2. Use the common ratio to find the 12th term:
- The 12th term is calculated as a12 = a1 * r^(12-1) = a1 * r^11
- Given that a1 * r^2 = 27, we can find a1:
- a1 = 27 / r^2 = 27 / (1.7632)^2 ≈ 27 / 3.1111 ≈ 8.6686 (approx.)
- Now, calculate the 12th term:
- a12 = 8.6686 * (1.7632)^11 ≈ 8.6686 * 160.4777 ≈ 1390.35 (approx.)
Therefore, the 12th term in the geometric sequence is approximately 1390.35.
