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 and the common ratio r = 4 .
Apply the formula for the nth term of a geometric sequence: a n = a 1 r n − 1 .
Substitute n = 7 , a 1 = 2 , and r = 4 into the formula: a 7 = 2 × 4 7 − 1 = 2 × 4 6 .
Calculate a 7 = 2 × 4096 = 8192 , so the 7th term is 8192 .
Explanation
Identifying the Problem We are given a geometric sequence and asked to find the 7th term. Let's first identify the key characteristics of this sequence.
Finding the First Term and Common Ratio The given geometric sequence is 2 , 8 , 32 , 128 , … . The first term, denoted as a 1 , is 2. To find the common ratio, r , we can divide any term by its preceding term. For example, r = 2 8 = 4 . We can verify this with other terms: 8 32 = 4 and 32 128 = 4 . So, the common ratio is indeed 4.
Stating 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. In our case, we want to find the 7th term, so n = 7 .
Calculating the 7th Term Now, we substitute the values we found into the formula: a 7 = 2 × 4 7 − 1 = 2 × 4 6 . We know that 4 6 = 4096 , so a 7 = 2 × 4096 = 8192 .
Final Answer Therefore, 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 amounts at the end of each year form a geometric sequence. Understanding geometric sequences helps you predict future values in these scenarios.
To find the 7th term in the geometric sequence 2, 8, 32, 128, ... we identify the first term as 2 and the common ratio as 4. Using the nth term formula, we calculate the 7th term to be 8192. Thus, a 7 = 8192 .
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