relatively simple sequence

[Dmitry]

Can somebody please explain how to derive that the explicit formula for
-1, 5, -9, 13, -17, ...

is a sub n = [(-1)^n](4n-3), n >= 1; or a sub n = [(-1)^(n+1)](4n+1)

I do not understand how it can be done

Thanks ahead of time!
Dmitry.

[gmalivuk]

The only difference there is whether you consider the first element (-1) to be a₀ or a₁.

[Dmitry]

I understand the part of (-1)^n and (-1)^(n+1)

what i do not understand is, what follows that.

why 4n-3, and why 4n+1

[Sizik]

Taking the absolute value of everything, the series is 1, 5, 9, 13, 17, 21, etc.. If you take the differences between successive elements, you get 4, 4, 4, 4, etc., which tells you that the series can be expressed as 4*0 + 1, 4*1 + 1, 4*2 + 1, etc. An easy way to express this would be a_n = 4n + 1, having the first element, 1, as a₀. If you wanted 1 to be a₁ instead, then you would have a_n = 4(n-1) + 1 = 4n + 1 - 4 = 4n - 3. Afterwards, just add in the (-1)ⁿ or (-1)ⁿ⁺¹, depending on which way you chose to describe the series.

[Dmitry]

Alright that cleared it up!

Thanks =]