Find the 6th term of this geometric sequence.
[tex]3,-12,48, \ldots[/tex]
[tex]a_6=[?][/tex]
Answer (2)
Determine the common ratio by dividing the second term by the first term: r = 3 − 12 = − 4 .
Apply the formula for the nth term of a geometric sequence: a n = a 1 n − 1 .
Substitute a 1 = 3 , r = − 4 , and n = 6 into the formula: a 6 = 3 ( − 4 ) 6 − 1 = 3 ( − 4 ) 5 .
Calculate a 6 : a 6 = 3 ( − 1024 ) = − 3072 . The 6th term is − 3072 .
Explanation
Understanding the Problem We are given a geometric sequence and asked to find the 6th term. The sequence starts with 3, -12, 48, ...
Finding the Common Ratio To find the 6th term, we first need to determine the common ratio, which is the ratio between consecutive terms. We can find the common ratio r by dividing the second term by the first term: r = 3 − 12 = − 4
Using the General Formula Now that we have the common ratio, we can use the formula for the nth term of a geometric sequence: a n = a 1 n − 1 where a 1 is the first term and r is the common ratio. In our case, a 1 = 3 and r = − 4 .
Finding the 6th Term We want to find the 6th term, so we set n = 6 : a 6 = 3 6 − 1 = 3 5 = 3 ( − 4 ) 5
Calculating the 6th Term Now we calculate ( − 4 ) 5 : ( − 4 ) 5 = − 1024 So, a 6 = 3 ( − 1024 ) = − 3072
Final Answer Therefore, the 6th term of the geometric sequence is -3072.
Examples
Geometric sequences are useful in many real-world applications, such as calculating compound interest, population growth, and radioactive decay. 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 scenarios.
The 6th term of the geometric sequence 3, -12, 48, ... is found to be -3072. This is calculated using the common ratio of -4 and applying the formula for the nth term of a geometric sequence. Therefore, the answer is -3072.
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