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Chapter 12: Problem 7

Find the \(n\)th term of the geometric sequence with given first term \(a\) and common ratio \(r .\) What is the fourth term? $$ a=\frac{5}{2}, \quad r=-\frac{1}{2} $$

Short Answer

Expert verified
The fourth term of the sequence is \(-\frac{5}{16}\).

Step by step solution

01

Identify the Formula

The general formula to find the nth term of a geometric sequence is given by \[ a_n = a imes r^{(n-1)} \] where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.
02

Input the Values

Substitute the given values into the formula, where \( a = \frac{5}{2} \) and \( r = -\frac{1}{2} \):\[ a_n = \left( \frac{5}{2} \right) \times \left( -\frac{1}{2} \right)^{n-1} \]
03

Calculate the Fourth Term

To find the fourth term \( a_4 \), set \( n = 4 \):\[ a_4 = \left( \frac{5}{2} \right) \times \left( -\frac{1}{2} \right)^{3} \]
04

Simplify the Expression

Calculate \( \left( -\frac{1}{2} \right)^{3} \):\[ \left( -\frac{1}{2} \right)^{3} = -\frac{1}{8} \]Now substitute back:\[ a_4 = \left( \frac{5}{2} \right) \times \left( -\frac{1}{8} \right) \]
05

Final Calculation

Multiply the fractions:\[ a_4 = \frac{5}{2} \times -\frac{1}{8} = -\frac{5}{16} \]Thus, the fourth term is \(-\frac{5}{16}\).

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Key Concepts

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

The nth Term Formula
When dealing with geometric sequences, one key aspect is finding any term in the sequence using the nth term formula. Understanding this formula gives you the power to explore the entire sequence without having to list out every single term. A geometric sequence is one where each term is developed by multiplying the previous term by a constant value, known as the common ratio. The nth term formula is crucial here and it is given by: \[ a_n = a \times r^{(n-1)} \] **Breaking It Down:**
  • \( a_n \) represents the value of the nth term.
  • \( a \) is the first term of the sequence.
  • \( r \) is the common ratio - the factor by which we multiply each term to get the next one.
  • \( n-1 \) denotes that we start counting from the first term when applying the ratio \( r \).
This formula condenses the process into something simple and efficient, allowing you to find a term situated at any position in a sequence.
Understanding the Common Ratio
The common ratio is the backbone of a geometric sequence. It is the value that you consistently multiply by to go from one term to the next. Understanding the common ratio helps in predicting how the sequence will progress. **Key Insights:**
  • The common ratio \( r \) can be any real number: positive, negative, or even a fraction.
  • If \( r > 1 \), the terms increase exponentially. It makes the sequence grow larger as you move along.
  • If \( 0 < r < 1 \), the terms decrease, leading to a sequence that shrinks towards zero.
  • If \( r = -1 \), the sequence will alternate; positive then negative, so on.
  • With \( |r| < 1 \), as time or terms go on, it leads the terms toward zero slowly.
For example, in our solution, the common ratio was \(-\frac{1}{2}\). Let鈥檚 see its consequence: it causes the terms to flip signs alternately and shrink since its absolute value is less than one. Recognizing the pattern determined by the common ratio allows for intuition about how the sequence behaves without crunching all the numbers.
Discovering the First Term
The first term \( a \) of a geometric sequence lays the initial ground for what the sequence will look like. Every geometric sequence begins with a specific first term that acts as the seed from which the rest of the sequence grows. This term fundamentally affects the sequence鈥檚 progression in combination with the common ratio. **Important Points to Consider:**
  • \( a \) is crucial, as it provides the base value which is multiplied by the common ratio to find subsequent terms.
  • A larger first term can lead to larger terms in the sequence faster, especially with a ratio greater than one.
  • In our example, the first term is \(\frac{5}{2}\). Being positive, it will influence the initial direction and magnitude for the rest of the sequence before factoring in the common ratio.
Thus, while the sequence progresses from one term to another using the common ratio, the first term is the cornerstone from which this progression springs.

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Most popular questions from this chapter

Find the term containing \(y^{3}\) in the expansion of \((\sqrt{2}+y)^{12}\)

Funding an Annuity 55 -year-old man deposits \(\$ 50,000\) to fund an annuity with an insurance company. The money will be invested at 8\(\%\) per year, compounded semi- annually. He is to draw semiannual payments until he reaches age \(65 .\) What is the amount of each payment?

Amortizing a Mortgage When they bought their house, John and Mary took out a \(\$ 90,000\) mortgage at 9\(\%\) interest, repayable monthly over 30 years. Their payment is \(\$ 724.17\) per month (check this using the formula in the text). The bank gave them an amortization schedule, which is a table showing how much of each payment is interest, how much goes toward the principal, and the remaining principal after each payment. The table below shows the first few entries in the amortization schedule. $$ \begin{array}{|c|c|c|c|c|}\hline \text { Payment } & {\text { Total }} & {\text { Interest }} & {\text { Principal }} & {\text { Remaining }} \\\ {\text { number }} & {\text { payment }} & {\text { payment }} & {\text { payment }} & {\text { principal }} \\ \hline 1 & {724.17} & {675.00} & {49.54} & {89,950.83} \\ {2} & {724.17} & {674.63} & {49.54} & {89,901.29} \\\ {3} & {724.17} & {674.26} & {49.91} & {89,851.38} \\ {4} & {724.17} & {673.89} & {50.28} & {89,801.10} \\ \hline\end{array} $$ After 10 years they have made 120 payments and are wondering how much they still owe, but they have lost the amortization schedule. (a) How much do John and Mary still owe on their mortgage? [Hint: The remaining balance is the present value of the 240 remaining payments. (b) How much of their next payment is interest and how much goes toward the principal? [Hint: Since 9\(\% \div\) \(12=0.75 \%\) , they must pay 0.75\(\%\) of the remaining principal in interest each month.

Find the sum of the infinite geometric series. $$ -\frac{100}{9}+\frac{10}{3}-1+\frac{3}{10}-\cdots $$

Find the first three terms in the expansion of $$ \left(x+\frac{1}{x}\right)^{40} $$
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