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Chapter 14: Problem 20

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 \(a_{1}=4, r=-2\).

Short Answer

Expert verified
-8192

Step by step solution

01

Identify Formula

For a geometric sequence, the \(n^{th}\) term (\(a_{n}\)) can be found using the formula \(a_{n} = a_{1} \cdot r^{(n-1)}\), where \(a_{1}\) is the first term and \(r\) is the common ratio.
02

Substitute Given Values

Here, we are given \(a_{1} = 4\), \(r = -2\), and we need to find \(a_{12}\). So, \(n = 12\). Substituting these values into the formula, we get \(a_{12} = 4 \cdot (-2)^{(12-1)}\).
03

Perform Calculation

Now, simplify the above expression to get the value of \(a_{12}\). Calculation: \(a_{12} = 4 \cdot (-2)^{(11)} = 4 \cdot -2048 = -8192\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

General Term Formula
The general term formula of a geometric sequence is like a magic key that helps unlock any term of the sequence. This formula allows us to calculate the term located at any position, known as the "nth" position. To achieve this, we use the formula: \(a_{n} = a_{1} \cdot r^{(n-1)}\). Here's what each part means:
  • \(a_{n}\): This is the "nth" term we are trying to find. For example, if you want to find the 12th term, "12" is "n".
  • \(a_{1}\): This is the first term of our sequence. It鈥檚 the starting point of our journey through the sequence.
  • \(r\): This is the common ratio, which we'll discuss more soon.
  • \(n-1\): This indicates the number of steps we take from the first term鈥攆rom \(a_{1}\) to \(a_{n}\).
The true power of this formula lies in its reliability. Once you understand it, you can navigate through any geometric sequence effortlessly.
nth Term Calculation
Once you have the general term formula, finding the "nth" term is a straightforward process. To calculate it, you just need to substitute the values you have into the formula. Let鈥檚 see how it works with an example.Given the first-term \(a_{1} = 4\) and the common ratio \(r = -2\), if we need to calculate \(a_{12}\), our formula setup would look like this: \(a_{12} = 4 \cdot (-2)^{11}\).
  • Substitute \(n = 12\) because we want the 12th term.
  • Insert the known value of the first term \(a_{1} = 4\).
  • Apply the common ratio: in our case, \(-2\).
In this example, plugging in and simplifying gives: \(a_{12} = 4 \cdot (-2048) = -8192\). Just like that, you've calculated the 12th term from this sequence!
Common Ratio
The common ratio in a geometric sequence is a crucial element. It determines how each term progresses from the one before. Essentially, it's the factor by which you multiply (or divide) one term to get the next. In a formula, this is represented by "\(r\)".
  • In any geometric sequence, the common ratio remains constant throughout.
  • It can be a positive or negative number, as well as a fraction.
For the example we worked with, \(r = -2\). This means each term is obtained by multiplying the previous term by \(-2\). Having a negative common ratio can lead to alternating signs in the sequence, introducing an exciting pattern.
Exponentiation in Sequences
Exponentiation plays a significant role in geometric sequences, particularly in influencing how sequences grow. When we raise the common ratio \(r\) to the power of \((n-1)\), we are determining how many times the ratio should be applied to reach the desired term. In our example, calculating \((-2)^{11}\) means applying the factor \(-2\) across 11 consecutive transformations:
  • Exponentiation calculates powers, so \((-2)^{11}\) essentially multiplies \(-2\) by itself 11 times.
  • This multiplication exponentially increases or decreases the value based on the size and sign of the common ratio.
Therefore, exponentiation allows for understanding how different terms are spaced and scaled in a geometric sequence. It's the mechanism behind the dynamic leaps between each term.

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Most popular questions from this chapter

Use the formula for the sum of the first n terms of a geometric sequence to solve. A pendulum swings through an arc of 20 inches. On each successive swing, the length of the arc is \(90 \%\) of the previous length. $$\begin{aligned}&20, \quad\quad 0.9(20), \quad0.9^{2}(20), \quad 0.9^{3}(20), \ldots\\\&\begin{array}{|c|c|c|c|}\hline \text { 1st } & \text { 2nd } & \text { 3rd } & \text { 4th } \\\\\text { swing } & \text { swing } & \text { swing } & \text { swing } \\\\\hline\end{array}\end{aligned}$$ After 10 swings, what is the total length of the distance the pendulum has swung? Round to the nearest hundredth of an inch.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If the \(n\) th term of a geometric sequence is \(a_{n}=3(0.5)^{n-1}\) the common ratio is \(\frac{1}{2}\).

Use the formula for the value of an annuity to solve Exercises. Round answers to the nearest dollar. At age \(25,\) to save for retirement, you decide to deposit \(\$ 50\) at the end of each month in an IRA that pays \(5.5 \%\) compounded monthly. a. How much will you have from the IRA when you retire at age \(65 ?\) b. Find the interest.

Use the formula for the sum of the first n terms of a geometric sequence to solve. A job pays a salary of \(\$ 24,000\) the first year. During the next 19 years, the salary increases by \(5 \%\) each year. What is the total lifetime salary over the 20 -year period? Round to the nearest dollar.

Use a graphing utility to graph the function. Determine the horizontal asymptote for the graph of \(f\) and discuss its relationship to the sum of the given series. Function $$f(x)=\frac{4\left[1-(0.6)^{x}\right]}{1-0.6}$$ Series $$\begin{array}{l}4+4(0.6)+4(0.6)^{2} \\\\+4(0.6)^{3}+\cdots\end{array}$$
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