Arithmetic Sequences Solutions and Guide

Arithmetic Sequences and General Term Formula

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is referred to as the common difference, denoted by $d$. The general term, or the $n$\text{th} term of an arithmetic sequence, can be determined using the explicit formula:

$a_n = a_1 + (n - 1)d$

In this formula, $a_n$ represents the value of the $n$\text{th} term, $a_1$ represents the first term of the sequence, $n$ represents the term number or position in the sequence, and $d$ represents the common difference between consecutive terms.

Worked Example 1: Finding an Explicit Term in a Simple Sequence

To find the $11$\text{th} term of the sequence $3, 4, 5, \dots$, first identify the key parameters from the given sequence. The first term $a_1$ is $3$. The common difference $d$ is calculated by subtracting the first term from the second term, giving $d = 4 - 3 = 1$. The target term position $n$ is $11$

Substituting these values into the explicit formula $a_n = a_1 + (n - 1)d$ gives:

a_{11} = 3 + (11 - 1) \n\times 1

Simplifying inside the parentheses yields:

$a_{11} = 3 + (10) \times 1$

Performing the multiplication:

$a_{11} = 3 + 10$

Adding the terms gives the final result:

$a_{11} = 13$

Worked Example 2: Calculating Higher-Order Terms

Consider an arithmetic sequence where the first term $a_1 = 17$ and the common difference $d = 4$. To compute the $20$\text{th} term ($a_{20}$), substitute the given parameters $a_1 = 17$, $d = 4$, and $n = 20$ into the standard arithmetic term equation.

The general formula $a_n = a_1 + (n - 1)d$ becomes:

$a_{20} = 17 + (20 - 1) \times 4$

Subtract $1$ from $20$ within the expression:

$a_{20} = 17 + (19) \times 4$

Multiply $19$ by $4$ to find the product:

$(19) \times 4 = 76$

Add this product to the initial term $17$:

$a_{20} = 17 + 76 = 93$

Thus, the $20$\text{th} term of the sequence is $93$

Worked Example 3: Evaluating Terms in Increasing Multiples

To determine the $42$\text{nd} term of the sequence $5, 10, 15, \dots$, determine the components of the sequence. The initial term $a_1$ is $5$. The common difference $d$ is calculated as $10 - 5 = 5$. The specified term number $n$ is $42$

Applying the formula $a_n = a_1 + (n - 1)d$ yields:

$a_{42} = 5 + (42 - 1) \times 5$

Subtract $1$ from $42$:

$a_{42} = 5 + (41) \times 5$

Multiply $41$ by $5$:

$(41) \times 5 = 205$

Add $5$ to $205$ to obtain the value of the $42$\text{nd} term:

$a_{42} = 5 + 205 = 210$

Worked Example 4: Finding the Term Number Given the Value

In problems where the term value $a_n$ is given as $-395$, the first term $a_1 = 5$, and the common difference $d = 5$, the objective is to solve for the term position $n$

Set up the formula $a_n = a_1 + (n - 1)d$ with the known values:

$-395 = 5 + (n - 1) \times 5$

Subtract $5$ from both sides of the equation:

$-395 - 5 = (n - 1) \times 5$

$-400 = (n - 1) \times 5$

Divide both sides by the common difference $5$:

$\frac{-400}{5} = n - 1$

$-80 = n - 1$

Add $1$ to both sides to solve for $n$:

$n = -80 + 1 = -79$

Worked Example 5: Determining the Common Difference

When given the first term $a_1 = 5$, a term value $a_n = 17$, and the term number $n = 7$, the task is to find the common difference $d$

Using the formula $a_n = a_1 + (n - 1)d$, substitute $a_7 = 17$, $a_1 = 5$, and $n = 7$:

$17 = 5 + (7 - 1)d$

Simplify the term expression in parentheses:

$17 = 5 + 6d$

Subtract $5$ from both sides of the equation:

$17 - 5 = 6d$

$12 = 6d$

Divide by $6$ to isolate $d$:

$d = \frac{12}{6} = 2$

The common difference for the given sequence parameters is $2$