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Chapter 9: Problem 2

Fill in the blanks. The \( n \)th term of an arithmetic sequence has the form ________.

Short Answer

Expert verified
The nth term of an arithmetic sequence has the form \( a_n = a_1 + (n-1)d \).

Step by step solution

01

Understanding the arithmetic sequence

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is a constant, referred to as the common difference. This property must be held for all consecutive pairs of numbers in the sequence.
02

Forming the nth term formula for an arithmetic sequence

For an arithmetic sequence, there exists a specific formula that expresses the nth term. Given the first term of the sequence \( a_1 \) and the common difference \( d \), this formula takes the form: \( a_n = a_1 + (n-1)d \) where \( n \) refers to the term being calculated.
03

Example of finding the nth term in an arithmetic sequence

Suppose we have an arithmetic sequence with the first term \( a_1 = 2 \), and a common difference \( d = 3 \). If we wanted to find the 5th term \( a_5 \), we would substitute into the formula as follows: \( a_5 = a_1 + (5 - 1)d = 2 + (4)3 = 14 \). Thus, the 5th term is 14.

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Key Concepts

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

Arithmetic Sequence
An arithmetic sequence is like a regularly spaced row of numbers, where each number is a fixed distance away from the one before it. Imagine standing on the steps of a staircase; each step you take upward is exactly the same height as the last. In the world of arithmetic sequences, that 'height,' or gap between numbers, is known as the common difference.

Take a sequence like 2, 4, 6, 8, and so on. It's easy to see that we're adding 2 each time to get to the next number. The sequence continues indefinitely in both increasing or decreasing direction, adhering to this established pattern. It's a foundational concept in mathematics and underpins many aspects of algebra and calculus.
Common Difference
The term 'common difference' might sound complex, but it's just a technical term for something very simple: the distance between neighboring numbers in an arithmetic sequence. It's 'common' because it's the same for every pair of consecutive numbers in the sequence. Whether the difference is positive, like in the sequence 3, 6, 9, where we add 3 each time, or negative, such as in the sequence 10, 5, 0, (subtracting 5 each time), it's this consistent step that defines an arithmetic sequence. Mathematics is all about patterns, and the common difference is the key to unlocking a number of problems related to arithmetic sequences.
Sequence Formula
The sequence formula for an arithmetic sequence is like a recipe that tells us how to cook up any term in the sequence. Given the first term, known as a1, and that all-important common difference d, the nth term of the sequence is found using the formula:
\[ a_n = a_1 + (n-1)d \]
Think of n as the 'ingredient' that varies depending on which term you want. If you want the 10th term, plug in 10 for n. Want the 50th term? Use 50 for n. It's a straightforward way to jump directly to any term in the sequence without having to write out the whole thing. Very handy for calculations and understanding patterns within sequences!

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Most popular questions from this chapter

In Exercises 91 - 94, prove the identity. \( _nC_n = _nC_0 \)

In Exercises 73 - 76, find the number of diagonals of the polygon. (A line segment connecting any two non adjacent vertices is called a diagonal of the polygon.) Decagon (\( 10 \) sides)

In Exercises 91 - 94, prove the identity. \( _nC_r = \dfrac{_nP_r}{r!} \)

In Exercises 53 - 60, the sample spaces are large and you should use the counting principles discussed in Section 9.6. One card is selected at random from an ordinary deck of \( 52 \) playing cards. Find the probabilities that (a) the card is an even-numbered card,(b) the card is a heart or a diamond, and (c) the card is a nine or a face card.

In Exercises 53 - 60, the sample spaces are large and you should use the counting principles discussed in Section 9.6. A shipment of \( 12 \) microwave ovens contains three defective units. A vending company has ordered four of these units, and because each is identically packaged, the selection will be random. What are the probabilities that (a) all four units are good,(b) exactly two units are good, and (c) at least two units are good?
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