The first term of a sequence is 4 and the common ratio is -2. What is the formula for the sequence based on form number
$a_n=-2 \cdot(4)^{n-1}$
a. $-4 \cdot(-2)^2-1$
$a_n-(4=-2)^{n-1}$
$a_n=4 \cdot(-2)^n$
Given that a sequence is defined by the formula $a_n=4 \cdot(-2)^{n-1}$, find the 20th form.
Answer (1)
The formula for the nth term of the geometric sequence is a n = 4 ⋅ ( − 2 ) n − 1 .
To find the 20th term, substitute n = 20 into the formula: a 20 = 4 ⋅ ( − 2 ) 19 .
Simplify the expression: a 20 = − 4 ⋅ 2 19 = − 2 21 .
Calculate the final value: a 20 = − 2097152 .
Explanation
Understanding the Problem We are given a geometric sequence with the first term a 1 = 4 and a common ratio r = − 2 . We want to find the formula for the sequence and then find the 20th term.
Finding the Formula The general formula for the nth term of a geometric sequence is given by a n = a 1 ⋅ r n − 1 . In our case, a 1 = 4 and r = − 2 . Therefore, the formula for the sequence is a n = 4 ⋅ ( − 2 ) n − 1 .
Finding the 20th Term Now we want to find the 20th term of the sequence, which means we need to find a 20 . We substitute n = 20 into the formula we found: a 20 = 4 ⋅ ( − 2 ) 20 − 1 = 4 c d o t ( − 2 ) 19 .
Simplifying the Expression Since ( − 2 ) 19 = − 2 19 , we have a 20 = 4 ⋅ ( − 2 19 ) = − 4 ⋅ 2 19 . We can rewrite this as a 20 = − 2 2 ⋅ 2 19 = − 2 21 .
Calculating the Final Value Now we calculate 2 21 . The result of the operation is 2 21 = 2097152 . Therefore, a 20 = − 2097152 .
Final Answer Thus, the 20th term of the sequence is -2097152.
Examples
Geometric sequences are useful in many real-world applications, such as calculating compound interest, modeling population growth or decay, 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 can help you make informed financial decisions and predict future outcomes in various scenarios.
