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Chapter 15: Problem 8

In Exercises \(8-11\), find the \(5^{\text {th }}, 10^{\text {th }}, n^{\text {th }}\) term of the arithmetic sequences. $$ 4,6,8, \ldots $$

Short Answer

Expert verified
Answer: The 5th term is 12, the 10th term is 22, and the nth term is given by the formula \(a_n = 4 + (n-1) \cdot 2\).

Step by step solution

01

Find the first term and common difference

The sequence begins with the first term \(a_1 = 4\). The common difference can be found by subtracting the first term from the second term, which gives \(d = 6 - 4 = 2\).
02

Use the arithmetic sequence formula

We use the arithmetic sequence formula: \(a_n = a_1 + (n-1)d\). We know that \(a_1 = 4\) and \(d = 2\). Now we'll substitute these values into the formula for the 5th, 10th, and nth terms.
03

Find the 5th term

To find the 5th term, we need to find the value of \(a_5\). Using the formula, we have: \(a_5 = a_1 + (5-1)d = 4 + (5-1) \cdot 2 = 4 + 8 = 12\)
04

Find the 10th term

To find the 10th term, we need to find the value of \(a_{10}\). Using the formula, we have: \(a_{10} = a_1 + (10-1)d = 4 + (10-1) \cdot 2 = 4 + 18 = 22\)
05

Find the nth term

To find the nth term, we need to find the value of \(a_n\). Using the formula, we have: \(a_n = a_1 + (n-1)d = 4 + (n-1) \cdot 2\) As a result, the \(5^{\text{th}}\) term is 12, the \(10^{\text{th}}\) term is 22, and the \(n^{\text{th}}\) term of the arithmetic sequence is \(a_n = 4 + (n-1) \cdot 2\).

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Key Concepts

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

First Term
Understanding the **first term** of an arithmetic sequence is like identifying the starting point of a journey. In mathematical sequences, this first term is often denoted by the symbol \(a_1\). For any arithmetic sequence, knowing this initial value is crucial, as it sets the tone for the rest of the sequence.
In the example we've been given, the sequence starts with the first term \(a_1 = 4\). This means that 4 is the number from which every subsequent term builds upon. Imagine lining up a series of numbers where 4 is in front, and every other number joins behind it according to a specific rule.
The **key takeaway** is, without this first term, it would be impossible to determine the rest of the sequence. It is the anchor to which everything else is attached.
  • The first term establishes the sequence.
  • Denoted as \(a_1\) in formulas.
  • It's essential for calculating other terms in the sequence.
Common Difference
In an arithmetic sequence, the **common difference** is the constant amount added (or subtracted) to each term to get to the next one. This is another vital part of understanding and working with these sequences.
To figure out the common difference, you just need to subtract the first term from the second term. For example, in our sequence \(4, 6, 8, \ldots\), the common difference \(d\) is calculated by \(6 - 4 = 2\). Essentially, this means each number is 2 more than the previous one.
This common difference maintains uniformity in the sequence. Whether calculating the 5th term, 10th term, or any \(n^{\text{th}}\) term, this difference is key to ensuring consistent progression from one term to the next.
  • The common difference helps determine the pattern of the sequence.
  • Calculated by subtracting the first term from the second term.
  • Represented as \(d\) in formulas.
Arithmetic Sequence Formula
The **arithmetic sequence formula** is a tool used to find any term in the sequence without listing all the previous numbers. The formula is given by: \[ a_n = a_1 + (n-1) \times d \] where \(a_n\) represents the \(n^{\text{th}}\) term you want to find, \(a_1\) is the first term, and \(d\) is the common difference.
Let's explore why this formula is so powerful. It simplifies the process of finding terms in the sequence by providing a direct computation method. For instance, if you want the 5th term, just plug \(n = 5\) into the formula. Using our sequence \(4, 6, 8, \ldots\), where \(a_1 = 4\) and \(d = 2\), the formula becomes \(a_n = 4 + (n-1) \times 2\).
Whether you need the 5th term, 22, or the 150th, this formula works efficiently, eliminating the need to calculate every term individually up to that point.
  • Used for finding any term in the sequence.
  • Includes first term, common difference, and term position.
  • Streamlines calculations for terms far into the sequence.

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Most popular questions from this chapter

For each of the four series (a)-(d), identify which have the same number of terms and which have the same value. (a) \(\sum_{i=1}^{20} 3\) (b) \(\sum_{j=1}^{20}(3 j)\) (c) \(0+3+6 \ldots+60\) (d) \(60+57+54+\cdots+6+3\)

In Exercises 2-4, complete the tables with the terms of the arithmetic series \(a_{1}, a_{2}, \ldots, a_{n},\) and the sequence of partial sums, \(S_{1}, S_{2}, \ldots, S_{n} .\) State the values of \(a_{1}\) and \(d\) where \(a_{n}=a_{1}+(n-1) d\) The table for the sequence \(3,8,13,18, \ldots\) is $$ \begin{array}{c|c|c|c|c|c|c|c|c} \hline n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline a_{n} & 3 & 8 & 13 & 18 & & & & \\ \hline S_{n} & 3 & 11 & 24 & 42 & & & & \\ \hline \end{array} $$

Two brothers are dividing M\&Ms between themselves. The older one gives one to his younger brother and takes one for himself. He gives another to his younger brother and takes two for himself. He gives a third one to younger brother and takes three for himself, and so on. (a) On the \(n^{\text {th }}\) round, how many M\&Ms does the older boy give his younger brother? How many does he take himself? (b) After \(n\) rounds, how many M\&Ms does his brother have? How many does the older boy have?

Evaluate the sums in Problems 5-12 using the formula for the sum of an arithmetic series. $$ -4.01-4.02-4.03-\cdots-4.35 $$

Figure 15.3 shows the quantity of the drug atenolol in the body as a function of time, with the first dose at time \(t=0 .\) Atenolol is taken in \(50 \mathrm{mg}\) doses once \(a\) day to lower blood pressure. (a) If the half-life of atenolol in the body is 6 hours, what percentage of the atenolol \(^{9}\) present at the start of a 24 -hour period is still there at the end? (b) Find expressions for the quantities \(Q_{0}, Q_{1}, Q_{2},\) \(Q_{3}, \ldots,\) and \(Q_{n}\) shown in Figure \(15.3 .\) Write the expression for \(Q_{n}\) (c) Find expressions for the quantities \(P_{1}, P_{2}, P_{3}, \ldots,\) and \(P_{n}\) shown in Figure \(15.3 .\) Write the expression for \(P_{n}\)
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