# Finding a formula for the sequence

• Sep 24th 2011, 02:23 PM
l flipboi l
Finding a formula for the sequence
Can someone help me with this?

I need to find the formula for this sequence?

{10,-2,10,-2,...}

Generally, what's the best technique for trying to find the formula?

I know the difference between term 1 and term 2 is -12, which alternates signs after that.

Thanks,
• Sep 24th 2011, 02:31 PM
Plato
Re: Finding a formula for the sequence
Quote:

Originally Posted by l flipboi l
I need to find the formula for this sequence?
{10,-2,10,-2,...}

Notation $\displaystyle \mod(n,2)$ means n modulo 2.
The sequence is $\displaystyle a_n=10\cdot\text{mod}(n,2)-2\cdot\text{mod}(n+1,2)$ where $\displaystyle n=1,2,3,\cdots$.
• Sep 24th 2011, 03:11 PM
Soroban
Re: Finding a formula for the sequence
Hello, l flipboi l

Quote:

Find the formula for this sequence: .$\displaystyle 10,-2,10,-2,\:\hdots$

When we have a sequence of alternating terms, I use the "blinker" function.

. . . . $\displaystyle f(n) \;=\;\frac{1-(\text{-}1)^n}{2}A + \frac{1+(\text{-}1)^n}{2}B$

Note that: .$\displaystyle f(n) \;=\;\begin{Bmatrix}A & \text{for odd }n \\ \\[-4mm] B & \text{for even }n \end{array}$

One function would be: .$\displaystyle f(n) \;=\;\frac{1-(\text{-}1)^n}{2}(10) + \frac{1+(\text{-}1)^n}{2}(-2)$

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The function can be severely simplified.

$\displaystyle \frac{1-(\text{-}1)^n}{2}(10) + \frac{1+(\text{-}1)^n}{2}(-2)$

. . $\displaystyle =\;\left[1-(\text{-}1)^n\right]\cdot5 \,-\, \left[1 + (\text{-}1)^n\right]$

. . $\displaystyle =\;5 - (\text{-}1)^n\!\cdot\!5 - 1 - (\text{-}1)^n$

. . $\displaystyle =\;\left[5-1\right] - \left[5+1\right](\text{-}1)^n$

. . $\displaystyle =\;4 - 6(\text{-}1)^n$

This form is more readily understood:
. . Take the "middle number" and alternately add and subtract 6.

• Sep 24th 2011, 07:00 PM
l flipboi l
Re: Finding a formula for the sequence
Thanks! I was so stuck at trying to add/subtract 12 to a number. I'm pretty new at finding a formula for the sequence.

Any tips on what should I look out for? or is the solution merely a trial and error?
• Sep 24th 2011, 07:37 PM
TKHunny
Re: Finding a formula for the sequence
No, not trial and error. You should proceed with your best judgment. You should also be aware that there is NO single best answer. Given a finite number of values, there are infinitely many solutions. If you create one you can justify, anyone grading your paper should not DARE to mark it wrong. There is no "simplest" version, as some will claim. "simplest" cannot be defined in this context.

The question is fundamentally flawed. It absolutely should not say "the formula". It should say "a formula".