/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 16 Find the indicated term of the a... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 16

Find the indicated term of the arithmetic sequence with first term, \(a_{1},\) and common difference, \(d .\) Find \(a_{16}\) when \(a_{1}=9, d=2\)

Short Answer

Expert verified
The 16th term of the arithmetic sequence is 39.

Step by step solution

01

Identify the given values

The first term of the sequence \(a_1 = 9\) and the common difference \(d = 2\) are given. The term we need to find, \(a_{n}\), is \(a_{16}\).
02

Use the arithmetic sequence formula

The formula for finding the nth term of an arithmetic sequence is \(a_n = a_1 + (n-1) * d\). In this exercise, the values to input are: \(n = 16\), \(a_1 = 9\), and \(d = 2\).
03

Calculate the 16th term

Input the values into the formula to calculate the 16th term in the sequence. So, \(a_{16} = 9 + (16-1) * 2 = 9 + 30 = 39

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Key Concepts

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

Nth Term of an Arithmetic Sequence
The concept of the 'nth term' is a fundamental aspect of arithmetic sequences, which are simply lists of numbers that are created by adding the same value each time, known as the common difference. The 'nth term' refers to the value of any specific term in the sequence when its position is known.

For example, if we're trying to find the 16th term of a sequence (the 'nth term'), and we know the first term and the common difference, we can predict the value of the term in that position without listing all preceding terms. This becomes incredibly practical with long sequences where writing each term individually would be impractical.

Understanding how to calculate the nth term is key in a variety of mathematical problems, including finding specific values in sequences, understanding series, and even predicting patterns. It's this predictive power that makes arithmetic sequences a critical topic in algebra and beyond.
Common Difference in Arithmetic Sequence
The 'common difference' is what sets arithmetic sequences apart from other number patterns. It is the difference between any two consecutive terms, and it remains constant throughout the sequence. Identifying the common difference is paramount because it helps us understand the sequence's rate of change.

In most problems dealing with arithmetic sequences, you will either be given the common difference or asked to calculate it. To find it, simply subtract one term in the sequence from the next one. For instance, in the sequence where the first term is 9 and the second term is 11, the common difference is 2, since 11 - 9 = 2. This common difference is then used to find any term in the sequence using the arithmetic sequence formula. Knowing the common difference allows us to build or deconstruct the entire sequence.
Arithmetic Sequence Formula
Arithmetic sequences follow a simple formula: \(a_n = a_1 + (n-1) * d\) where \(a_n\) is the nth term you're looking to find, \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference. This formula provides a straightforward method to calculate any term within the sequence without having to construct the entire sequence.

When you're faced with an arithmetic sequence problem, the first step is usually to identify these key components鈥攖he first term and the common difference. Once these pieces of information are clear, applying the formula becomes a matter of simple arithmetic. The beauty of the formula lies in its ability to save time and improve accuracy, as it eliminates the potential errors that could occur when calculating each term manually.

Always remember that understanding and applying this formula correctly is essential for solving arithmetic sequence questions efficiently, whether on homework, tests, or in practical, real-world scenarios.

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Most popular questions from this chapter

Here are two ways of investing \(\$ 30,000\) for 20 years. $$ \begin{array}{ccc} {\text { Lump-Sum Deposit }} & {\text { Rate }} & {\text { Time }} \\ {\$ 30,000} & {5 \% \text { compounded }} & {20 \text { years }} \\ {} & {\text { annually }} \end{array} $$ $$ \begin{array}{ll} {\text { Periodic Deposits }} & {\text { Rate } \quad \text { Time }} \\ {\$ 1500 \text { at the end }} & {5 \% \text { compounded } 20 \text { years }} \\ {\text { of each year }} & {\text { annually }} \end{array} $$ After 20 years, how much more will you have from the lump-sum investment than from the annuity?

Mega Millions is a multi-state lottery played in most U.S. states. As of this writing, the top cash prize was \(\$ 656\) million, going to three lucky winners in three states. Players pick five different numbers from 1 to 56 and one number from 1 to \(46 .\) Use this information to solve Exercises \(27-30 .\) Express all probabilities as fractions. A player wins a minimum award of \(\$ 150\) by correctly matching three numbers drawn from white balls \((1 \text { through } 56\) ) and matching the number on the gold Mega Ball \((1 \text { through } 46)\) What is the probability of winning this consolation prize?

Suppose that it is a drawing in which the Powerball jackpot is promised to exceed \(\$ 700\) million. If a person purchases \(292,201,338\) tickets at \(\$ 2\) per ticket (all possible combinations), isn't this a guarantee of winning the jackpot? Because the probability in this situation is \(1,\) what's wrong with doing this?

The table shows the population of California for 2000 and \(2010,\) with estimates given by the U.S. Census Bureau for 2001 through 2009 \(\begin{array}{lllllll}\hline \text { Year } & {2000} & {2001} & {2002} & {2003} & {2004} & {2005} \\ \hline \text { Population } & {33.87} & {34.21} & {34.55} & {34.90} & {35.25} & {35.60} \\ \hline\end{array}\) \(\begin{array}{llllll}{\text { Year }} & {2006} & {2007} & {2008} & {2009} & {2010} \\ {\text { Population }} & {36.00} & {36.36} & {36.72} & {37.09} & {37.25}\end{array}\) a. Divide the population for each year by the population in the preceding year. Round to two decimal places and show that California has a population increase that is approximately geometric. b. Write the general term of the geometric sequence modeling California's population, in millions, \(n\) years after 1999 c. Use your model from part (b) to project California's population, in millions, for the year \(2020 .\) Round to two decimal places.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I used a formula to find the sum of the infinite geometric series \(3+1+\frac{1}{3}+\frac{1}{9}+\cdots\) and then checked my answer by actually adding all the terms.
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