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Chapter 8: Problem 32

In Exercises 31 and 32, describe and correct the error in writing a rule for the \(n\)th term of the geometric sequence for which \(a_2=48\) and \(r=6\). $$ \begin{aligned} a_n &=r\left(a_1\right)^{n-1} \\ 48 &=6\left(a_1\right)^{2-1} \\ 8 &=a_1 \\ a_n &=6(8)^{n-1} \end{aligned} $$

Short Answer

Expert verified
The corrected rule for the nth term of the geometric sequence is \(a_n = 8 \cdot 6^{n-1}\). The error originated in calculating the first term \(a_1\).

Step by step solution

01

Identify the Error

Looking at the given equations, the student correctly identified that \(a_2 = 48\) and \(r = 6\). The error comes from the calculation of the first term \(a_1\). The student has calculated the first term as \(48 / 6 = 8\), but this is incorrect since \(a_2 = a_1r\). So, we should have \(a_1 = a_2 / r\).
02

Calculate Correct First Term

Let's correct that mistake by reassessing how we get \(a_1\). We know that \(a_2 = a_1 \cdot r\). Hence, to get \(a_1\) we should divide \(a_2\) by \(r\), so \(a_1 = 48 / 6 = 8\).
03

Form Correct Rule for nth Term

Now that we have the correct \(a_1\), we can form the geometric series' nth term, which gives us \(a_n = a_1 \cdot r^{n-1} = 8 \cdot 6^{n-1}\).

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Key Concepts

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

nth Term Formula
The nth term of a geometric sequence gives us a way to find any term in the sequence without listing all preceding terms. It's like finding the page in a book directly instead of flipping through the pages.

The formula for the nth term of a geometric sequence is:
  • \(a_n = a_1 imes r^{n-1}\)
Here, \(a_n\) is the nth term, \(a_1\) is the first term, and \(r\) is the common ratio.

To use this formula effectively:
  • Understand that \(a_1\) is the starting number of your sequence.
  • \(r\) is how much you multiply each term by to get the next term.
  • \(n\) is the position of the term in the sequence you are looking for.
The formula saves time and effort, letting you calculate any term directly without listing them all. Remember, always confirm that you have used the correct values for the first term and the common ratio to avoid errors.
Common Ratio
In a geometric sequence, the common ratio is a crucial concept. It represents the factor by which we multiply each term in the sequence to get the next term.

To find the common ratio \(r\):
  • Take any term in the sequence, say \(a_n\), and divide it by the previous term, \(a_{n-1}\).
  • This can be expressed as \(r = \frac{a_n}{a_{n-1}}\).
The common ratio helps determine the nature of the sequence:
  • If \(|r| > 1\), the sequence grows or shrinks rapidly.
  • If \(0 < |r| < 1\), the sequence converges towards zero.
  • If \(r = -1\), the sequence will oscillate between positive and negative values.
The choice of \(r\) impacts the characteristics of the geometric sequence significantly.
First Term Calculation
Often, the first term of a geometric sequence is given, but when it's not, or if partial data is provided, you'll need to calculate it. For example, in our original error exercise, determining \(a_1\) was crucial to formulating the correct nth term formula.

To find the first term \(a_1\) when you know the second term \(a_2\) and the common ratio \(r\):
  • Use the relationship \(a_2 = a_1 \times r\).
  • Rearrange to find \(a_1 = \frac{a_2}{r}\).
This calculation ensures you have the correct starting point:
  • It's vital for forming the entire sequence accurately.
  • Errors here can lead to every other term in the sequence being incorrect.
Getting \(a_1\) right aligns your sequence setup correctly, ensuring all subsequent calculations are spot on.

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Most popular questions from this chapter

Tell whether the function represents exponential growth or exponential decay. Then graph the function. \(y=e^{0.25 x}\)

The variables x and y vary inversely. Use the given values to write an equation relating x and y.Then fi nd y when x = 4. $$ x=2, y=9 $$

MAKING AN ARGUMENT Your friend believes the sum of a series doubles when the common difference of an arithmetic series is doubled and the first term and number of terms in the series remain unchanged. Is your friend correct? Explain your reasoning.

In Exercises 5鈥�12, tell whether the sequence is geometric. Explain your reasoning. \(2,4,6,8,10, \ldots\)

In 1202, the mathematician Leonardo Fibonacci wrote Liber Abaci, in which he proposed the following rabbit problem: Begin with a pair of newborn rabbits. When a pair of rabbits is two months old, the rabbits begin producing a new pair of rabbits each month. Assume none of the rabbits die. $$ \begin{array}{|l|l|l|l|l|l|l|} \hline \text { Month } & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \begin{array}{l} \text { Pairs at start } \\ \text { of month } \end{array} & 1 & 1 & 2 & 3 & 5 & 8 \\ \hline \end{array} $$ This problem produces a sequence called the Fibonacci sequence, which has both a recursive formula and an explicit formula as follows. $$ \text { Recursive: } a_1=1, a_2=1, a_n=a_{n-2}+a_{n-1} $$ $$ \text { Explicit: } f_n=\frac{1}{\sqrt{5}}\left(\frac{1+\sqrt{5}}{2}\right)^n-\frac{1}{\sqrt{5}}\left(\frac{1-\sqrt{5}}{2}\right)^n, n \geq 1 $$ Use each formula to determine how many rabbits there will be after one year. Justify your answers.
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