Today I am going to look at Geometric Progression, which is also known as G.P. Geometric Progression is a term used when a starting number, a, is reduced/increased by a fixed percentage or fraction. G.P. only involves the operation; division or multiplication. There is another topic similar to G.P. by increasing/reducing a start number, a, by a fixed value by means of operations; subtraction or addition. This is known as Arithmetic Progression also known as A.P. This topic will be taught in my next post.
G.P. is a similar form of gradual depreciation/appreciation and A.P. is a similar form of fixed depreciation/appreciation, but we will look at G.P. in this post.
As you have already known the depreciation formula from the previous post:
V x ((100 – d)/ 100)^n
The G.P. formula is almost exactly the same, it has the same principle but has different terms.
The formula for the G.P. is
ar^( n- 1)
Where ‘a’ is the beginning value (same as ‘V’ in the depreciation formula. Where ‘r’ is the common ratio (same as ((100- d)/100) in the depreciation formula) but there is a slight difference with the ‘n’.
G.P. really gives the same answer as the depreciation formula, but it is not the same ‘n’ as in the depreciation formula.
The ‘n’ in the depreciation formula tells you the amount of years as in (Dec 31st) not as in (Jan 1st). The ‘n’ in the G.P. formula tells you the amount of years as in (Jan 1st) hence it is (n – 1) in the G.P. formula so that it works out to be the exact value as in the depreciation formula.
The G.P doesn’t also workout percentages (depreciation/appreciation) it can also be used in different cases.
E.g. An amoeba (single cell) doubles every 30 minutes. How many cells are in the container after 3 hours?
This can be done manually………..>
3 hours = 6 x 30 minutes…..thus it doubles 6 times!
2^6 = 64
Start amount | time elapsed |
1 | 30 minutes |
2 | 1 hour |
4 | 3 x 30 minutes |
8 | 4 x 30 minutes |
16 | 5 x 30 minutes |
32 | 6 x 30 minutes |
64
So you started at 1 and ended with 64, hence there are 7 terms. (the ‘n’ in the G.P. formula gives the number of terms!)
Using the G.P. formula
ar^( n- 1)
a = 1 r = 2 and n = 7, we will get 64.
Questions like these are disguised in the S.A.T but with practice you will able to identify and use the G.P. formula efficiently. Well, that concludes G.P. ( you can try the depreciation question from the previous post using the G.P. method). The next topic is on Arithmetic Progression.
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