Find the eighth term of the geometric sequence, given the first term and common ratio.
[tex]
\begin{array}{c}
a_1=6 \text { and } r=-\frac{1}{3} \\
a_8=-\underline{[?]}
\end{array}
[/tex]
Answer (1)
Recall the formula for the nth term of a geometric sequence: a n = a 1 n − 1 .
Substitute the given values a 1 = 6 , r = − 3 1 , and n = 8 into the formula: a 8 = 6 ( − 3 1 ) 8 − 1 .
Calculate ( − 3 1 ) 7 = − 2187 1 .
Multiply by 6 and simplify: a 8 = 6 ( − 2187 1 ) = − 729 2 .
The eighth term of the geometric sequence is − 729 2 .
Explanation
Understanding the Problem We are given a geometric sequence with the first term a 1 = 6 and a common ratio r = − 3 1 . We want to find the eighth term, a 8 .
Recall the Formula The formula for the n th term of a geometric sequence is given by: a n = a 1 ⋅ r n − 1 where a 1 is the first term, r is the common ratio, and n is the term number.
Substitute the Values In our case, we have a 1 = 6 , r = − 3 1 , and we want to find a 8 , so n = 8 . Plugging these values into the formula, we get: a 8 = 6 ⋅ ( − 3 1 ) 8 − 1 = 6 ⋅ ( − 3 1 ) 7
Calculate the Power and Simplify Now we need to calculate ( − 3 1 ) 7 . Since the exponent is odd, the result will be negative: ( − 3 1 ) 7 = − 3 7 1 = − 2187 1 So, a 8 = 6 ⋅ ( − 2187 1 ) = − 2187 6 We can simplify the fraction by dividing both the numerator and the denominator by 3: a 8 = − 2187 ÷ 3 6 ÷ 3 = − 729 2
State the Answer Therefore, the eighth term of the geometric sequence is − 729 2 .
Examples
Geometric sequences are useful in many areas of mathematics and in real-world applications. For example, they can model the decay of radioactive substances, the growth of populations, or the calculation of compound interest. Understanding geometric sequences helps in predicting future values based on a constant ratio of change, which is essential in financial planning, scientific research, and engineering design. For instance, if you invest money with a fixed annual interest rate, the yearly balances form a geometric sequence, allowing you to project your investment's growth over time using the formula we applied here: a n = a 1 n − 1 .
