Find the $7^{\text {th }}$ term in the sequence $1,-3,9,-27, \ldots$
Hint: Write a formula to help you.
$1^{\text {st }}$ term - Common Ratio ${ }^{(\text {desired term }-1)}$
Remember to use the correct Order of Operations!
Answer (1)
Identify the first term a 1 = 1 and the common ratio r = − 3 .
Use the formula for the n th term of a geometric sequence: a n = a 1 n − 1 .
Substitute n = 7 , a 1 = 1 , and r = − 3 into the formula: a 7 = 1 ⋅ ( − 3 ) 7 − 1 = ( − 3 ) 6 .
Calculate ( − 3 ) 6 = 729 , so the 7 th term is 729 .
Explanation
Understanding the Problem We are given the sequence 1 , − 3 , 9 , − 27 , … and we want to find the 7 th term. We are also given the hint to use the formula: 1 st term ⋅ Common Ratio ( desired term − 1 ) . This looks like a geometric sequence.
Finding the First Term and Common Ratio First, we need to identify the first term, a 1 , and the common ratio, r . The first term is clearly a 1 = 1 . To find the common ratio, we can divide any term by its preceding term. For example, 1 − 3 = − 3 , − 3 9 = − 3 , 9 − 27 = − 3 . So the common ratio is r = − 3 .
Applying the Formula Now we can use the formula for the n th term of a geometric sequence: a n = a 1 n − 1 . We want to find the 7 th term, so n = 7 . Plugging in the values we found, we have a 7 = 1 ⋅ ( − 3 ) 7 − 1 = 1 ⋅ ( − 3 ) 6 .
Calculating the 7th Term Now we calculate ( − 3 ) 6 . Since the exponent is even, the result will be positive. ( − 3 ) 6 = ( − 3 ) × ( − 3 ) × ( − 3 ) × ( − 3 ) × ( − 3 ) × ( − 3 ) = 9 × 9 × 9 = 81 × 9 = 729 . Therefore, a 7 = 1 ⋅ 729 = 729 .
Final Answer The 7 th term in the sequence is 729 .
Examples
Geometric sequences are useful in many real-world scenarios, such as calculating compound interest, modeling population growth, and determining the depreciation of assets. For example, if you invest $1000 in an account that earns 5% interest compounded annually, the amount of money you have each year forms a geometric sequence. Understanding geometric sequences helps you predict future values in these types of situations.
