Problem 11
Question
Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of each sequence with the given first term, \(a_{1},\) and common ratio, \(r .\) Find \(a_{12}\) when \(a_{1}=5, r=-2\)
Step-by-Step Solution
Verified Answer
So, the 12th term of the geometric sequence (\(a_{12}\)) is -10240.
1Step 1: Identify the known variables
The first term (\(a_{1}\)) is given as 5 and the common ratio (\(r\)) is -2. We are asked to find the 12th term, so \(n=12\).
2Step 2: Apply the general formula
We apply the formula \(a_{n} = a_{1} * r^(n-1)\) using the given values. Plugging in the values gives: \(a_{12} = 5 * (-2)^(12-1)\).
3Step 3: Solve the calculation
Solve the calculation, remembering that any number to an odd power will keep its sign. \(a_{12} = 5 * (-2)^{11}\) will then become \(a_{12} = 5 * -2048\).
4Step 4: Compute the answer
Final step is to multiply 5 by -2048 which gives \(a_{12} = -10240\).
Other exercises in this chapter
Problem 11
In Exercises \(11-30,\) use mathematical induction to prove that each statement is true for every positive integer \(n\) $$ 4+8+12+\dots+4 n=2 n(n+1) $$
View solution Problem 11
In Exercises \(9-16,\) use the formula for \(_{n} C_{r}\) to evaluate each expression. $$ _{11} C_{4} $$
View solution Problem 11
Write the first four terms of each sequence whose general term is given. $$a_{n}=\frac{(-1)^{n+1}}{2^{n}-1}$$
View solution Problem 12
Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (x+3 y)^{3} $$
View solution