Question

What is an explicit formula for an arithmetic sequence with a common difference of three and whos second term is nine?

Answer 1

• First term =9-3=6
• Second term=9
• Common difference=3

So

• a=6
• d=3

Explicit formula

`\\ \rm\Rrightarrow a_n=a+(n-1)d`

`\\ \rm\Rrightarrow a_n=6+3(n-1)`

Answer 2

`\sf a_n=3n+3`

Explanation:

An explicit formula for an arithmetic sequence allows you to find the nth term of the sequence.

A recursive formula for an arithmetic sequence allows you to find the nth term of the sequence provided you know the value of the previous term in the sequence.

Explicit formula

`\sf a_n=a+(n-1)d`

where:

• 
  `\sf a_n` is the nth term
• a is the first term
• n is the number of the term
• d is the common difference

Given:

• 
  `\sf a_2=9`
• d = 3
• n = 2

Substituting these values into the formula to find a:

`\implies \sf 9=a+(2-1)3`

`\implies \sf 9=a+3`

`\implies \sf a=6`

Therefore the formula is:

`\implies \sf a_n=6+(n-1)3`

`\implies \sf a_n=6+3n-3`

`\implies \sf a_n=3n+3`