Find the ninth term of the geometric sequence, given the first term and common ratio.
[tex]
\begin{array}{c}
a_1=1 \text { and } r=-\frac{1}{3} \\
a_9=\frac{[?]}{\square}
\end{array}
[/tex]
Answer (1)
The problem provides the first term and common ratio of a geometric sequence.
We recall the formula for the nth term of a geometric sequence: a n = a 1 n − 1 .
We substitute the given values into the formula to find the ninth term: a 9 = 1 ( − 3 1 ) 9 − 1 .
We simplify the expression to obtain the final answer: 6561 1 .
Explanation
Understanding the Problem We are given a geometric sequence with the first term a 1 = 1 and a common ratio r = − 3 1 . We want to find the ninth term, a 9 .
Recall the Formula The formula for the n th term of a geometric sequence is given by a n = a 1 ⋅ r n − 1 .
Substitute the Values In our case, we have a 1 = 1 , r = − 3 1 , and n = 9 . Substituting these values into the formula, we get: a 9 = 1 ⋅ ( − 3 1 ) 9 − 1 = ( − 3 1 ) 8
Simplify the Expression Now, we simplify the expression: ( − 3 1 ) 8 = 3 8 ( − 1 ) 8 = 3 8 1 We calculate 3 8 : 3 8 = 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 = 6561 Therefore, a 9 = 6561 1
State the Answer The ninth term of the geometric sequence is 6561 1 .
Examples
Geometric sequences are useful in calculating compound interest, population growth, and radioactive decay. For example, if you invest $1000 with an annual interest rate of 5% compounded annually, the amount you have each year forms a geometric sequence. Understanding geometric sequences helps in predicting future values in various financial and scientific applications.
