Things to expect with grade 10 Geometric Sequences

[wScott]

We're going to be starting them in a day or two, and I just wanted to know ahead of time what you guys might think we'll be learning with them, like formulae and that kind of stuff..

 

[HallsofIvy]

There's not a whole lot you have to know about "geometric sequences". A geometric sequence is one in which you go from one number to the next by always multiplying by the same thing.

Example: 3, 6, 12, 24, 48, 96,... The point is that 6/3= 2, 12/6= 2, 48/24= 2, 96/48= 2. In other words, you start with the number 3 and proceed to just keep multiplying by 2. If we start counting terms with n= 0 (some people start with n= 1) then the "n^th" term is 3(2)ⁿ or, in more general terms, arⁿ where a is the first term and a is the "common multiplier"j. If you start counting with n= 1, then the n^th term is ar^n-1:you have to subtract 1 to get back to 0.

Another nice property is this: suppose we add the terms (a geometric series rather than sequenc). For example if S= a+ ar+ ar²+ ar³, then S= a+ r(a+ ar+ ar²) where we've just lost one power. Okay, put it back in: S= a+ r(a+ ar+ ar²+ ar³- ar³. See how I added and subtracted the same thing? Now separate those: S= a+ r(a+ ar+ ar²+ ar³)+ ar⁴= a+ r(S)+ ar⁴ so
S- rS= S(1-r)= a+ ar⁴= a(1- r⁴). That is,
S(1-r)= a(1- r⁴) so S= a (1-r⁴)/(1- r).

More generally, the sum of the first n terms of a geometric sequence is
S_n= a(1-rⁿ⁺¹)/(1- r).

 

[wScott]

Well, it seems pretty simple to me, especially after the few arithmatic formulae we've learned over the past few days. Should be fun :)