# Find a formula that generates the following sequence

Find a formula which generates the following sequence.

$$15,20,25,30,35 \ldots$$

The answer is $5(n + 2)$

How? I know it comes from the formula $a_n = a_1 + (n - 1) d$, but I am not sure how they got the answer. Could someone please help me?

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The answer...**to what**? – DonAntonio Mar 13 '13 at 2:28
@Don, the answer to "find a formula that generates the following sequence". – Gerry Myerson Mar 13 '13 at 2:30
I was almost sure of that, @GerryMyerson, yet I think students above 4th grade must be thorough in the way they write and stop being sloppy. This is why I asked, but now it never minds. – DonAntonio Mar 13 '13 at 2:32

The difference between each of the terms is $5$, so we know that our formula should be:

$$a_n = 5n + c$$

Where $c$ is a constant term which we have not yet found.

We know that:

$$a_1 = 5 \cdot 1 + c$$ $$a_1 = 5 + c$$

But we are given $a_1 = 15$, and we have:

$$15 = 5 + c$$ $$10 = c$$

So we get:

$$a_n = 5n + 10$$

Factoring:

$$a_n = 5(n + 2)$$

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thank you very much!...i totally understood it here – Angela Mar 13 '13 at 2:41

$$15,20,25,30,35,\dots$$ They're all multiples of $5$, so divide by $5$ to get $$3,4,5,6,7\dots$$ Subtract $2$ to get $$1,2,3,4,5,\dots$$ but this is $n$. Add back the $2$, so $$3,4,5,6,7,\dots{\rm\ is\ }n+2$$ Multiply back the $5$, so $$15,20,25,30,35,\dots{\rm\ is\ }5(n+2)$$

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thank you for your help – Angela Mar 13 '13 at 2:42

This is an arithmetic progression, or arithmetic sequence: perhaps you'd like to read the linked entry to develop some tools for dissecting/constructing a sequence, and forming the term $a_n$.

You seemed to have a good start...

...so I think your main "sticking points" are algebraic, and grasping how the sequence can be analyzed to "deconstruct" and reconstruct as Gerry did so nicely.

Keep at it; the more progressions of this sort that you encounter and work with, the quicker you'll be able to "spot" what's going on!

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