Problem 16
Question
Write the first five terms of each arithmetic sequence with the given first term and common difference. $$a_{1}=-19, d=5$$
Step-by-Step Solution
Verified Answer
The first five terms of the arithmetic sequence with the given first term (\(a_1 = -19\)) and common difference (\(d = 5\)) are -19, -14, -9, -4, and 1.
1Step 1: Identify the first term and common difference
The given first term, \(a_1\), is -19, and the common difference, \(d\), is 5. We will use these values in the arithmetic sequence formula to find the first five terms.
2Step 2: Find the 2nd term, \(a_2\)
To find the 2nd term, we will use the arithmetic sequence formula with \(n = 2\):
\(a_2 = a_1 + (2-1)d = -19 + (1)(5) = -19 + 5 = -14\)
The second term, \(a_2\), is -14.
3Step 3: Find the 3rd term, \(a_3\)
To find the 3rd term, we will use the arithmetic sequence formula with \(n = 3\):
\(a_3 = a_1 + (3-1)d = -19 + (2)(5) = -19 + 10 = -9\)
The third term, \(a_3\), is -9.
4Step 4: Find the 4th term, \(a_4\)
To find the 4th term, we will use the arithmetic sequence formula with \(n = 4\):
\(a_4 = a_1 + (4-1)d = -19 + (3)(5) = -19 + 15 = -4\)
The fourth term, \(a_4\), is -4.
5Step 5: Find the 5th term, \(a_5\)
To find the 5th term, we will use the arithmetic sequence formula with \(n = 5\):
\(a_5 = a_1 + (5-1)d = -19 + (4)(5) = -19 + 20 = 1\)
The fifth term, \(a_5\), is 1.
6Step 6: Write the first five terms
Now that we have found the first five terms, we can write them as a sequence:
\(-19, -14, -9, -4, 1\)
The first five terms of the arithmetic sequence with the given first term (\(a_1 = -19\)) and common difference (\(d = 5\)) are -19, -14, -9, -4, and 1.
Key Concepts
First TermCommon DifferenceSequence FormulaTerms
First Term
An arithmetic sequence always begins with the first term. It's the starting point of the sequence, and everything else is built upon it. In our example, the first term is denoted as \( a_1 \) and is given as -19.
Think of the first term as the foundation of a house. All other parts (or terms) of the sequence are constructed from it. This term is crucial because every other term in the sequence relies on the value of this first term and the common difference, which we will talk about next.
A strong understanding of the first term also helps in predicting the future terms of the sequence accurately.
Think of the first term as the foundation of a house. All other parts (or terms) of the sequence are constructed from it. This term is crucial because every other term in the sequence relies on the value of this first term and the common difference, which we will talk about next.
A strong understanding of the first term also helps in predicting the future terms of the sequence accurately.
Common Difference
In an arithmetic sequence, the common difference is the amount you'll add (or subtract if it's negative) to move from one term to the next. It's represented by the letter \( d \).
In our problem, this common difference is 5. This means that each term is 5 units larger than the term before it. Essentially, the common difference acts like a step you take from one number to the next in the sequence.
Understanding the common difference helps you see the relationship between consecutive terms and enables you to find other terms in the sequence quickly.
In our problem, this common difference is 5. This means that each term is 5 units larger than the term before it. Essentially, the common difference acts like a step you take from one number to the next in the sequence.
Understanding the common difference helps you see the relationship between consecutive terms and enables you to find other terms in the sequence quickly.
Sequence Formula
The sequence formula for arithmetic sequences is a tool that allows you to find any term in the sequence. It is written as:
\[ a_n = a_1 + (n-1) \cdot d \] where \( a_n \) is the term you're looking to find, \( a_1 \) is the first term, \( n \) is the position of the term in the sequence, and \( d \) is the common difference.This formula works by relying on both the first term and the common difference to calculate subsequent terms. For instance, to find the second term, you substitute \( n = 2 \) into the formula.
\[ a_n = a_1 + (n-1) \cdot d \] where \( a_n \) is the term you're looking to find, \( a_1 \) is the first term, \( n \) is the position of the term in the sequence, and \( d \) is the common difference.This formula works by relying on both the first term and the common difference to calculate subsequent terms. For instance, to find the second term, you substitute \( n = 2 \) into the formula.
Terms
In an arithmetic sequence, each individual number is called a "term." The terms are the numbers you get as you move through the sequence. In our example, the first five terms are -19, -14, -9, -4, and 1.
Understanding how terms are connected through the sequence formula and the common difference is fundamental in analyzing the sequence. Each term can be calculated by applying the common difference again and again, starting from the first term.
Listing out the terms helps provide a visual representation and lets you check if your calculations are correct. Regularly reviewing the steps to find these terms will strengthen your understanding of arithmetic sequences.
Understanding how terms are connected through the sequence formula and the common difference is fundamental in analyzing the sequence. Each term can be calculated by applying the common difference again and again, starting from the first term.
Listing out the terms helps provide a visual representation and lets you check if your calculations are correct. Regularly reviewing the steps to find these terms will strengthen your understanding of arithmetic sequences.
Other exercises in this chapter
Problem 16
Find the general term, \(a_{m}\) for each geometric sequence. Then, find the indicated term. $$a_{1}=3, r=8 ; a_{3}$$
View solution Problem 16
Given the general term of each sequence, find each of the following. \(a_{n}=3 n-11\) a) the first term of the sequence b) \(a_{7}\) c) the 32 nd term
View solution Problem 17
Evaluate each binomial coefficient. $$\left(\begin{array}{l}7 \\\3\end{array}\right)$$
View solution Problem 17
Find the general term, \(a_{m}\) for each geometric sequence. Then, find the indicated term. $$a_{1}=-1, r=3 ; a_{5}$$
View solution