[SOLVED] Series/Sequence Problem?

[ae4jm]

I'm trying to figure out a formula for this sequence problem, rather than doing this over and over 1,000 times. Does anyone have a clue for the formula to find this? I've pasted the info and also the answer.

Thanks for your time!

[slider142]

Write out the formula for a1, a2, and a3 without summing and find the pattern so that you can write a formula for an that does not refer to an-1. It should remind you of a simple type of sum you already know a shortcut for.

[HallsofIvy]

{an} is defined, recursively, by a1= 4, a_{n+1}= a_n+ 4n. Find a1000.

The first thing I would do is start calculating a few values (hoping I won't have to go up to 1000!).

a2= 4+ 4(1), a[sub]3[/sup]= 4+ 4(1)+ 4(2), a[sub]4[/sup]= 4+ 4(1)+ 4(2)+ 4(3).

Hmmm, looks to me like an= 4(1+ 2+ 3+ ...(n-1)) so a1000= 4(1+ 2+ 3+ ... + 999). Can you find 1+ 2+ 3+ ...+ 999? It's an arithmetic sequence with common difference 1. Or you could use "Euler's method".

[tiny-tim]

Another method: rewrite the equation as:

an+1 - an = 4n;
so an+2 - an+1 = 4(n+1).

Add them … what do you get … ? And then … ? :smile:

[ae4jm]

Gentlemen, I'm totally stuck. I sat here for the last 1.5 hrs and tried to figure this one out. I think that I'm making it too difficult. I've looked at the sequence all the way up through a[sub]6 and I'm just not seeing the formula.

I do know that to find the sum from 1 to 1000, would be (1000(1001))/2 or (n(n+1))/2. But I've tried finding the a[sub]n like this also...dang I'm making this tuff aren't I?

[tiny-tim]

Hi ae4jm! :smile:

Humour me … what's an+2 - an? … what's an+3 - an? :smile:

[ae4jm]

I got 12 and 24???

[tiny-tim]

Hi ae4jm! :smile:

Yes … that's a3 - a1 and a4 - a1.

But you'll get more of an idea of what's going on if you answer the general question:

what's an+2 - an? … what's an+3 - an? (in terms of n):smile:

[ae4jm]

I believe that the first one is 6-4=2 and 7-4=3, right?

[ae4jm]

does this look correct for
a_{n}=4(\frac{n(n-1)}{2})+4

[tiny-tim]

:biggrin: Woohoo! :biggrin:

And now, of course, simplify it to 2n(n - 1) + 4.

Any questions? :smile:

[ae4jm]

Thanks! That was a dandy, for me anyways! I'm sure this practice will help me recognize these sequences a little better in the future. Thanks for sticking in there with me.