Questions: Given a geometric sequence with a2=10 and a3=-30, find a6.

Solution Steps

To find \( a_6 \) in a geometric sequence, we need to determine the common ratio \( r \) using the given terms \( a_2 \) and \( a_3 \). Once \( r \) is found, we can use the formula for the \( n \)-th term of a geometric sequence, \( a_n = a_1 \cdot r^{n-1} \), to find \( a_6 \).

1. Use the relationship \( a_3 = a_2 \cdot r \) to find \( r \).
2. Use the formula for the \( n \)-th term to find \( a_6 \).

Given the terms of the geometric sequence \( a_2 = 10 \) and \( a_3 = -30 \), we can find the common ratio \( r \) using the relationship: \[ r = \frac{a_3}{a_2} = \frac{-30}{10} = -3.0 \]

To find the first term \( a_1 \), we use the formula for the second term of a geometric sequence: \[ a_2 = a_1 \cdot r^{2-1} \implies a_1 = \frac{a_2}{r} = \frac{10}{-3.0} = -3.333 \]

Using the formula for the \( n \)-th term of a geometric sequence, \( a_n = a_1 \cdot r^{n-1} \), we find \( a_6 \): \[ a_6 = a_1 \cdot r^{6-1} = -3.333 \cdot (-3.0)^5 = 810.0 \]

Final Answer