Find the 7th term of this geometric sequence.
[tex]
\begin{array}{c}
2,8,32,128, \ldots \
a_7=[?]
\end{array}
[/tex]
Answer (2)
Identify the first term: a 1 = 2 .
Calculate the common ratio: r = 4 .
Apply the formula for the nth term of a geometric sequence: a n = a 1 n − 1 .
Calculate the 7th term: a 7 = 2 4 6 = 8192 . The 7th term of the geometric sequence is 8192 .
Explanation
Identifying the Problem We are given a geometric sequence and asked to find the 7th term. The sequence is: 2 , 8 , 32 , 128 , …
Finding the First Term To find the 7th term, we first need to determine the first term and the common ratio of the sequence. The first term is clearly a 1 = 2 .
Calculating the Common Ratio To find the common ratio, we can divide any term by its preceding term. For example, r = 2 8 = 4 . We can verify this with another pair of terms: 8 32 = 4 and 32 128 = 4 . So the common ratio is r = 4 .
Stating the General Formula The formula for the nth term of a geometric sequence is given by: a n = a 1 ⋅ r n − 1 .
Applying the Formula for the 7th Term We want to find the 7th term, so we need to calculate a 7 . We have a 1 = 2 , r = 4 , and n = 7 . Substituting these values into the formula, we get: a 7 = 2 ⋅ 4 7 − 1 = 2 ⋅ 4 6
Calculating the 7th Term Now we calculate 4 6 = 4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 4 = 4096 . Therefore, a 7 = 2 ⋅ 4096 = 8192 .
Final Answer Thus, the 7th term of the geometric sequence is 8192.
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 7th term of the geometric sequence is found by determining the first term and the common ratio, then applying the geometric sequence formula. Using the first term a 1 = 2 and the common ratio r = 4 , we calculate a 7 = 2 ⋅ 4 6 = 8192 . Therefore, the final answer is 8192.
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