Find the twelfth term of the geometric sequence, given the first term and the common ratio.
[tex]
\begin{array}{c}
a_1=2 \text { and } r=-1 \\
a_{12}=[?]
\end{array}
[/tex]
Answer (1)
The formula for the nth term of a geometric sequence is a n = a 1 n − 1 .
Substitute a 1 = 2 , r = − 1 , and n = 12 into the formula: a 12 = 2 ( − 1 ) 12 − 1 .
Simplify the exponent: a 12 = 2 ( − 1 ) 11 .
Calculate the final answer: a 12 = 2 ( − 1 ) = − 2 .
The twelfth term of the geometric sequence is − 2 .
Explanation
Understanding the Problem We are given a geometric sequence with the first term a 1 = 2 and a common ratio r = − 1 . We want to find the twelfth term, a 12 .
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.
Substituting the Values In our case, we have a 1 = 2 , r = − 1 , and we want to find a 12 , so n = 12 . Substituting these values into the formula, we get: a 12 = 2 ⋅ ( − 1 ) 12 − 1 a 12 = 2 ⋅ ( − 1 ) 11
Calculating the Result Since ( − 1 ) raised to an odd power is − 1 , we have ( − 1 ) 11 = − 1 . Therefore, a 12 = 2 ⋅ ( − 1 ) = − 2
Final Answer Thus, the twelfth term of the geometric sequence is − 2 .
Examples
Geometric sequences are useful in many areas of mathematics and in real-world applications. For example, understanding geometric sequences can help calculate the depreciation of an asset over time, model population growth under certain conditions, or determine the total amount earned from an investment with a fixed interest rate compounded over several periods. In this case, if you start with an initial investment of $2 and each period it multiplies by -1, after 12 periods you would have -$2.
