Question

given the following arithmetic sequence: 9, 23, 37, 51, 65, ... what is the common difference? what is the value of the first term? write a simplified expression for the right half of the equation that can be used to find the nth term in the sequence. $a_n =$

Explanation:

Common Difference

Step1: Recall common difference formula

In an arithmetic sequence, the common difference \( d \) is found by subtracting the first term from the second term (or any term from the next term). So \( d = a_{n + 1}-a_{n} \).

Step2: Calculate using given terms

Take the second term \( 23 \) and subtract the first term \( 9 \): \( d=23 - 9=14 \). We can verify with other terms: \( 37-23 = 14 \), \( 51 - 37=14 \), etc.

Step1: Identify the first term

The arithmetic sequence is given as \( 9, 23, 37, 51, 65, \dots \). The first term \( a_{1} \) is the first number in the sequence.

Step1: Recall arithmetic sequence formula

The formula for the nth term of an arithmetic sequence is \( a_{n}=a_{1}+(n - 1)d \), where \( a_{1} \) is the first term and \( d \) is the common difference.

Step2: Substitute \( a_{1} \) and \( d \)

We know \( a_{1}=9 \) and \( d = 14 \). Substitute these values into the formula: \( a_{n}=9+(n - 1)\times14 \).

Step3: Simplify the expression

Expand \( (n - 1)\times14 \): \( 14n-14 \). Then add \( 9 \): \( a_{n}=14n-14 + 9=14n-5 \).

Answer:

14

Value of the First Term