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Chapter 9: Problem 19

Find a formula for the \(n\)th term of the sequence. The sequence \(0,3,8,15,24, \ldots\)

Short Answer

Expert verified
The nth term is \(a_n = n^2 - 1\).

Step by step solution

01

Identify the differences between consecutive terms

Examine the sequence: \(0, 3, 8, 15, 24, \ldots\). Calculate the differences between consecutive terms.\[3 - 0 = 3, \quad 8 - 3 = 5, \quad 15 - 8 = 7, \quad 24 - 15 = 9\]The differences are \(3, 5, 7, 9, \ldots\). The differences between these terms are constant at \(2\). This implies the sequence is quadratic.
02

Assume a quadratic form and solve for coefficients

Assume the nth term of the sequence can be expressed as a quadratic: \(a_n = an^2 + bn + c\). Using the known terms, we set up the equations:- When \(n=1\), \(a(1)^2 + b(1) + c = 0\)- When \(n=2\), \(a(2)^2 + b(2) + c = 3\)- When \(n=3\), \(a(3)^2 + b(3) + c = 8\)This simplifies to the system:\[\begin{align*}a + b + c &= 0 \4a + 2b + c &= 3 \9a + 3b + c &= 8\end{align*}\]
03

Solve the system of equations

Subtract the first equation from the second:\[(4a + 2b + c) - (a + b + c) = 3 - 0 \Rightarrow 3a + b = 3\]Subtract the second equation from the third:\[(9a + 3b + c) - (4a + 2b + c) = 8 - 3 \Rightarrow 5a + b = 5\]Next, subtract these results:\[(5a + b) - (3a + b) = 5 - 3 \Rightarrow 2a = 2 \Rightarrow a = 1\]Substitute \(a = 1\) into \(3a + b = 3\):\[3(1) + b = 3 \Rightarrow b = 0\]Finally, substitute \(a = 1\), \(b = 0\) into \(a + b + c = 0\):\[1 + 0 + c = 0 \Rightarrow c = -1\]
04

Write the formula for the nth term

From the coefficients determined, the nth term formula of the sequence is:\[a_n = n^2 - 1\]

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Key Concepts

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

nth term formula
When dealing with sequences, it's important to identify patterns to predict any term in the sequence by its position, denoted by \(n\). This prediction is done using the \(n\)th term formula. For quadratic sequences, this formula typically takes the form \(a_n = an^2 + bn + c\).

This formula helps in calculating any term's value without listing all previous ones. Consider the sequence: \(0, 3, 8, 15, 24, \ldots\). If you follow through by using the derived formula \(a_n = n^2 - 1\), without solving multiple steps, you can find any desired term.

For instance:
  • The first term (\(n=1\)): \(1^2 - 1 = 0\).
  • The third term (\(n=3\)): \(3^2 - 1 = 8\).
  • Here, the formula simplifies the process of checking and validating against the sequence's terms.
sequence differences
Identifying the differences between consecutive terms in a sequence is a powerful tool in understanding its behavior. For quadratic sequences, these differences tell us about the nature of the sequence.

Let's break it down: we have the sequence \(0, 3, 8, 15, 24, \ldots\) and we observe differences: \(3, 5, 7, 9, \ldots\). The differences between these differences, also known as the second difference, are constant at \(2\).

This constant second difference is a hallmark of a quadratic sequence and signals us that the sequence can be expressed in the form of \(a_n = an^2 + bn + c\). This step is crucial because it guides us towards the correct type of formula needed to express the \(n\)th term.
system of equations
When we suspect a sequence follows a quadratic pattern, we use the quadratic formula form \(a_n = an^2 + bn + c\) and derive a system of equations from known terms. Based on the exercise, substituting terms from \(n=1\), \(n=2\), and \(n=3\) gives equations:
  • \(a + b + c = 0\)
  • \(4a + 2b + c = 3\)
  • \(9a + 3b + c = 8\)
These equations allow us to find the coefficients \(a, b,\) and \(c\). To solve them, we compare and simplify:
  • Subtracting equations to eliminate unknowns
  • Solving sequentially until isolated values for \(a, b,\) and \(c\) are found
Through this method, we identify \(a = 1, b = 0,\) and \(c = -1\). Thus, the \(n\)th term formula becomes \(a_n = n^2 - 1\). Solving systems of equations is fundamental in deriving formulas from sequences.

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Most popular questions from this chapter

A patient takes a 300 mg tablet for the control of high blood pressure every morning at the same time. The concentration of the drug in the patient's system decays exponentially at a constant hourly rate of \(k=0.12\) a. How many milligrams of the drug are in the patient's system just before the second tablet is taken? Just before the third tablet is taken? b. In the long run, after taking the medication for at least six months, what quantity of drug is in the patient's body just before taking the next regularly scheduled morning tablet?

Pythagorean triples \(\quad\) A triple of positive integers \(a, b,\) and \(c\) is called a Pythagorean triple if \(a^{2}+b^{2}=c^{2} .\) Let \(a\) be an odd positive integer and let $$b=\left\lfloor\frac{a^{2}}{2}\right\rfloor \quad \text { and } \quad c=\left\lceil\frac{a^{2}}{2}\right\rceil$$ be, respectively, the integer floor and ceiling for \(a^{2} / 2\). a. Show that \(a^{2}+b^{2}=c^{2} .\) (Hint: Let \(a=2 n+1\) and express \(b \text { and } c \text { in terms of } n .)\) b. By direct calculation, or by appealing to the accompanying figure, find $$\lim _{a \rightarrow \infty} \frac{\left\lfloor\frac{a^{2}}{2}\right\rfloor}{\left\lceil\frac{a^{2}}{2}\right\rceil}.$$

Determine if the sequence is monotonic and if it is bounded. $$a_{n}=\frac{2^{n} 3^{n}}{n !}$$

Find the values of \(x\) for which the given geometric series converges. Also, find the sum of the series (as a function of \(x\) ) for those values of \(x .\) $$\sum_{n=0}^{\infty} \sin ^{n} x$$

It is not yet known whether the series $$ \sum_{n=1}^{\infty} \frac{1}{n^{3} \sin ^{2} n} $$converges or diverges. Use a CAS to explore the behavior of the series by performing the following steps. a. Define the sequence of partial sums $$ s_{k}=\sum_{n=1}^{k} \frac{1}{n^{3} \sin ^{2} n} $$ What happens when you try to find the limit of \(s_{k}\) as \(k \rightarrow \infty ?\) Does your CAS find a closed form answer for this limit? b. Plot the first 100 points \(\left(k, s_{k}\right)\) for the sequence of partial sums. Do they appear to converge? What would you estimate the limit to be? c. Next plot the first 200 points \(\left(k, s_{k}\right) .\) Discuss the behavior in your own words. d. Plot the first 400 points \(\left(k, s_{k}\right) .\) What happens when \(k=355 ?\) Calculate the number \(355 / 113 .\) Explain from you calculation what happened at \(k=355 .\) For what values of \(k\) would you guess this behavior might occur again?
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