Find the 7th term of this geometric sequence.
[tex]
5, 10, 20, 40, \ldots
a_7=[?][/tex]
Answer (2)
Determine the common ratio: r = 5 10 = 2 .
Apply the formula for the nth term of a geometric sequence: a n = a 1 n − 1 .
Substitute a 1 = 5 , r = 2 , and n = 7 into the formula: a 7 = 5 2 7 − 1 = 5 2 6 .
Calculate the 7th term: a 7 = 564 = 320 , so the final answer is 320 .
Explanation
Problem Analysis We are given a geometric sequence and asked to find the 7th term. First, we need to determine the common ratio of the sequence.
Finding the Common Ratio To find the common ratio, we divide any term by its preceding term. Let's divide the second term by the first term: r = 5 10 = 2 So, the common ratio is 2.
General Formula The general formula for the nth term of a geometric sequence is given by: a n = a 1 n − 1 where a 1 is the first term, r is the common ratio, and n is the term number.
Calculating the 7th Term In our case, a 1 = 5 , r = 2 , and we want to find the 7th term, so n = 7 . Substituting these values into the formula, we get: a 7 = 5 × 2 7 − 1 = 5 × 2 6 = 5 × 64 = 320
Final Answer Therefore, the 7th term of the geometric sequence is 320.
Examples
Geometric sequences are useful in many real-world scenarios, such as calculating compound interest, population growth, and radioactive decay. For example, if you invest $5000 in an account that earns 8% 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.
The 7th term of the geometric sequence is 320. This is found by determining the common ratio as 2 and using the formula for the nth term of a geometric sequence. The calculation shows that the result is obtained by multiplying the first term by the common ratio raised to the power of (n-1).
;
