the first term of an arithmetic sequence is 42. the rule (a_{n}=a_{n - 1}+8) can be used to find the next…

the first term of an arithmetic sequence is 42. the rule (a_{n}=a_{n - 1}+8) can be used to find the next term of the sequence. explain how to write the explicit rule for the arithmetic sequence from the given information.

the first term of an arithmetic sequence is 42. the rule (a_{n}=a_{n - 1}+8) can be used to find the next term of the sequence. explain how to write the explicit rule for the arithmetic sequence from the given information.

Answer

Explanation:

Step1: Identify first - term and common - difference

The first term $a_1 = 42$. The rule $a_n=a_{n - 1}+8$ shows the common - difference $d = 8$.

Step2: Recall explicit formula for arithmetic sequence

The explicit formula for an arithmetic sequence is $a_n=a_1+(n - 1)d$.

Step3: Substitute values into formula

Substitute $a_1 = 42$ and $d = 8$ into the formula: $a_n=42+(n - 1)\times8$.

Step4: Simplify the formula

Expand $(n - 1)\times8$ to get $8n-8$. Then $a_n=42 + 8n-8=8n + 34$.

Answer:

The explicit rule for the arithmetic sequence is $a_n=8n + 34$, which is found by using the first - term $a_1 = 42$ and the common - difference $d = 8$ in the formula $a_n=a_1+(n - 1)d$ and then simplifying.