Question

What is the formula for the sequence `1, 3, 7, 9,...` ?

Answer 1

`n+(n-1)`

(assuming that the sequence ie 1,3,5,7,9)

Explanation:

`"n of " 2=3`
`"n of " 3=5`
`"n of " 4=7`
`"n of " 5=9`

Basically you just work out how different `a` is from `n` and adjust to suit. In this case, the pattern is the `n` plus what the previous `n` was: `n-1`.

Checking my idea of the pattern:
If `n = 1`, then a = `1+(1-1)=1+0=1` which is correct.

If `n = 7`, then a = `5+(5-1) = 5 + 4 = 9` which is also correct.

If `n = k`, then a = `k +(k-1) = 2k-1` which when k is substituted in is the same as above `(2*5=10)`

Answer 2

This sequence can be matched by:

`a_n = 3n-5/2-(-1)^n/2`

or by:

`a_n = -2/3n^3+5n^2-25/3n+5`

Explanation:

Assuming that the question is correct as given, there are several way to match the sequence:

`1, 3, 7, 9`

with a formula.

For example, if we add `(-1)^n/2` to each element, then we get the sequence:

`1/2, 7/2, 13/2, 19/2`

which is an arithmetic sequence with common difference `3`.

Hence we can write a formula for the given sequence:

`a_n = 3n-5/2-(-1)^n/2`

Alternatively, we could use the method of differences to match the sequence with a polynomial...

Write down the given sequence:

`color(blue)(1), 3, 7, 9`

Write down the sequence of differences between each consecutive pair of terms:

`color(purple)(2), 4, 2`

Write down the sequence of differences of those differences:

`color(brown)(2), -2`

Write down the sequence of differences of those differences:

`color(green)(-4)`

Having arrived at a constant sequence (albeit of just one element), we can use the initial term of each of these sequences as coefficients to give us a direct formula:

`a_n = color(blue)(1)/(0!) + color(purple)(2)/(1!)(n-1) + color(brown)(2)/(2!)(n-1)(n-2) + color(green)(-4)/(3!)(n-1)(n-2)(n-3)`

`color(white)(a_n) = 1+2n-2+n^2-3n+2-2/3n^3+4n^2-22/3n+4`

`color(white)(a_n) = -2/3n^3+5n^2-25/3n+5`