Question

Tell whether the sequence is arithmetic. Justify your answer. If the sequence is arithmetic, write a recursive

formula and an explicit formula to represent it.

3, 3.25, 3.5, 3.75,...
​

Answer 1

The explicit formula

`a_n = 3 +0.25 (n-1)`

The recursive formula

`a_1 = 3` for
`n=1`

`a_n = a_ {(n-1)} +0.25` if
`n>1`

Explanation:

If a sequence is arithmetical then the difference between any of its consecutive terms will be constant

3, 3.25, 3.5, 3.75,

`3.25-3 = 0.25\\\\3.5-3.25 = 0.25\\\\3.75 -3.5 = 0.25`

The difference between the consecutive terms remains constant so the sequence is arithmetic.

The explicit formula for an arithmetic sequence is:

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

Where d is the constant difference between the terms.

`d = 0.25`

`a_1` is the first term of the sequence.

`a_1 = 3`

So

`a_n = 3 +0.25 (n-1)`

Finally, the recursive formula is:

`a_1 = 3\\\\a_n = a_ {(n-1)} +0.25`