Question

Write a recursive formula for each explicit formula

and find the first term of the sequence.

38. an = 108 -n

40.) a = 35 + 52n​

Answer 1

38.
`a_1=107,\ a_n=a_(n-1)-1`

40.
`a_1=87,\ a_n=a_(n-1)+52`

Explanation:

38. Given explicit formula:

`a_n=108-n`

When
`n=1,` then
`a_1=108-1=107`

When
`n=2,` then
`a_2=108-2=106`

When
`n=3,` then
`a_3=108-3=105`

When
`n=4,` then
`a_4=108-4=104`

and so on

You can see that each next number is 1 less than previous number, so the recursive formula is

`a_n=a_(n-1)-1`

40. Given explicit formula:

`a_n=35+52n`

When
`n=1,` then
`a_1=35+52\cdot 1=87`

When
`n=2,` then
`a_2=35+52\cdot 2=139`

When
`n=3,` then
`a_3=35+52\cdot 3=191`

When
`n=4,` then
`a_4=35+52\cdot 4=243`

and so on

You can see that each next number is 52 more than previous number, so the recursive formula is

`a_n=a_(n-1)+52`