Geometric Series(es?)

- Hi, I’m ‘a’, the first term in a geometric series
- You may remember me from such series as the arithmetic series from Core 1
- The difference here is that instead of adding something to get from one term to the next, you multiply by something
- That something is the common ratio, ‘r’
- Thus, the series goes a, ar, ar^2, ar^3…
- So the nth term, if you will, is ar^(n-1)
- Sometimes you’ll be given a and r and asked to work out terms
- Other times you’ll be given some terms and asked to work out a and r
- Other times you’ll be given a or r and asked to work out the other, given a term or two
- Simultaneous equations are sometimes involved (solving by substitution, or even elimination)
- Hopefully the Core 2 examiners won’t get any more inventive than that, though

To Infinity (no further)

- If the common ratio is greater than 1, no sum to infinity for you. Go away.
- Wait, come back. Is it less than -1? If so, go away.
- If you’re still here, your common ratio is between (not including) -1 and 1
- That means it’s a convergent function, so a sum to infinity is possible
- Otherwise, it’s a divergent function, in which case the sum to infinity would literally be infinity, so it’s not worth calculating
- Convergent series get closer and closer and closer and closer (you get the picture) to a number
- You can find out what number your series is aiming at using the formula:
- Sum to infinity = a / (1 – r)
- It’s the first term divided by… 1 minus the common ratio
- And it only works for convergent series! (Ones which get smaller each time)

Proving the Sum to n

- Actually, I can’t be bothered to prove it here (mainly because to do so in ASCII would be rather difficult indeed)
- You do seem to need to be able to prove the formula for the sum of a geometric series to n, though, so it might be worth memorizing
- Anyway, what you get is this… (and it is in the Core 2 formula booklet)
- Sum to n = (a * (1 – r^n)) / (1 – r)… or something like that
- It’s pretty easy to use. You may have to use it in an inequality, too, just as a heads up.

###### Related articles

- Logarithms and Exponential Stuff – AS Maths Revision – Core Mathematics (C2) (mattg99.wordpress.com)
- The Binomial Expansion – AS Maths Revision – Core Mathematics (C2) (mattg99.wordpress.com)
- Factor, Remainder Theorems and Algebraic Long Division – AS Maths Revision – Core Mathematics (C2) (mattg99.wordpress.com)

Advertisements