ah got it. I meant the former ; ) thank you very much that helped : ) keep up the good work! I know that I’ll keep on trying to get this!

]]>Ah – do you mean “why is a geometric progression interesting/useful”, or “how did you derive the formula for a geometric progression?”

If the former, then popular applications in finance include the calculation of compounded or annualized sums where there is some cumulative interest at play. The multiplier is the interest rate, the constant the principal sum. Or you could think about the number of children in successive generations if there are, for example, 2.4 children per family…

If you want to know how to derive the formula, then you can use a very similar approach to the one you used for the arithmetic progression.

Write out the series longhand (I’m using ^ to mean “raise to the power”). This is often a good way to start to look at an algorithm for a problem – write out the answer you want in a longhand form – first using actual results, then seeing if there’s some formula you can use to derive each term. In this case, it is:

S = ar^m + ar^(m+1) + ar^(m+2) + … + ar^(n-1) + ar^n

Now, we’re looking to see if there’s some way we can cancel out most of the terms. In this case of an arithmetic series, we just subtracted the two series. In this case, we’ve got a multiplier to deal with, so we try multiplying both sides by r

Sr = ar^(m+1) + ar^(m+2) + ar^(m+3) + … + ar^n + ar^n+1

Now, if you subtract the second equation from the first, you end up with

S – Sr = ar^m – ar^(n+1)

The left hand side is easy. Look at the right hand sides and notice that the only term from the first equation not in the second is that initial ar^m, and the only term in the second not in the first is that final ar^(n+1).

You can rearrange this as:

S(1-r) = a(r^m – r^(n+1))

and dividing through by 1-r, you get

S = a(r^m – r^(n+1)) / (1-r)

Note that there was a typo in the original text here! Of all the thousands of people who have read the article (including the proofers!) no-one noticed. I’ve corrected it…

]]>in the 1 exercise there is an explanation as to why that algorithm is interesting, why is that not the case for the 2nd exercise I really want to understand this but it’s driving me nuts. ]]>

Yes, it should. And now it is as if it always had been :-)

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