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Chapter 10: Problem 20

Find a formula for the \(n\) th term of the sequence. The sequence \(-3,-2,-1,0,1, \ldots\)

Short Answer

Expert verified
The \(n\)th term of the sequence is \(n - 4\).

Step by step solution

01

Observe the Sequence

The given sequence is \(-3, -2, -1, 0, 1, \ldots\). Look at the list of numbers to identify a pattern or rule.
02

Identify the Pattern

The sequence increases by 1 with each term. The first term is \(-3\).
03

Establish a Formula

To find the \(n\)th term of the sequence, note that the first term can be written as \(-3 + 1(0)\), the second term as \(-3 + 1(1)\), the third term as \(-3 + 1(2)\), and so on. The pattern follows: \(-3 + (n-1)\times1\).
04

Simplify the Formula

Simplify the expression for the general term: \(-3 + (n-1) = n - 4\). Thus, the formula for the \(n\)th term is \(n - 4\).

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Key Concepts

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

Sequence Patterns
Recognizing patterns in sequences is crucial for understanding how they function. Look closely at the sequence:
  • -3, -2, -1, 0, 1, ...
Here, each number increases by 1 as we move from left to right. This consistent interval between the terms is a hallmark of an arithmetic sequence. In arithmetic sequences, the difference between consecutive terms remains constant. Hence, it's straightforward to predict the value of future terms. For this sequence, the difference, also known as the common difference, is 1. Identifying this pattern helps us establish the formula that describes the sequence.
Formula Derivation
Deriving a formula for sequences involves translating the pattern into a mathematical expression. In our sequence:
  • First term (\(a_1\)): -3
  • Common difference (\(d\)): 1
To find the formula for the \(n\)th term, we use the formula for arithmetic sequences: \(a_n = a_1 + (n-1)d\). Substituting the values:- \(a_1 = -3\)- \(d = 1\)The formula becomes:\[a_n = -3 + (n-1) \times 1\]Simplifying gives:\[a_n = -3 + n - 1 = n - 4\]This simplified expression \(n - 4\) represents the \(n\)th term for this specific sequence.
Mathematical Sequences
Mathematical sequences are lists of numbers following a particular rule. They are a fundamental concept in mathematics. Each term in a sequence is derived from a specific pattern or formula. Sequences can be finite or infinite, depending on the context.
  • Arithmetic sequences have a constant difference between terms.
  • Geometric sequences have a constant ratio between terms.
Understanding sequence types is essential for developing formulas to calculate terms. In our example, we focused on arithmetic sequences, perfect for modeling simple linear progressions. Mastering sequences helps in various math and real-life applications, from finance to physics.

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Most popular questions from this chapter

Does the series $$ \sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n^{2}}\right) $$ converge or diverge? Justify your answer.

The first term of a sequence is \(x_{1}=1 .\) Each succeeding term is the sum of all those that come before it: \(x_{n+1}=x_{1}+x_{2}+\cdots+x_{n}\). Write out enough early terms of the sequence to deduce a general formula for \(x_{n}\) that holds for \(n \geq 2\).

a. Series for \(\sinh ^{-1} x \quad\) Find the first four nonzero terms of the Taylor series for $$ \sinh ^{-1} x=\int_{0}^{x} \frac{d t}{\sqrt{1+t^{2}}} $$ b. Use the first three terms of the series in part (a) to estimate \(\sinh ^{-1} 0.25 .\) Give an upper bound for the magnitude of the estimation error.

Use a geometric series to represent each of the given functions as a power series about \(x=0,\) and find their intervals of convergence. a. \(f(x)=\frac{5}{3-x}\) b. \(\quad g(x)=\frac{3}{x-2}\)

Use a CAS to perform the following steps for the sequences. a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit \(L\) ? b. If the sequence converges, find an integer \(N\) such that \(\left|a_{n}-L\right| \leq 0.01\) for \(n \geq N .\) How far in the sequence do you have to get for the terms to lie within 0.0001 of \(L ?\) \(a_{1}=1, \quad a_{n+1}=a_{n}+(-2)^{n}\)
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