The formula works for both expressions. See my notes below.
sum ( r^k, k from 0 to n) is (1 - r^(n+1)) / ( 1 - r) is the same as
sum ( r^k, k from 0 to n-1) is (1 - r^n)) / ( 1 - r).
Not sure which way your book states it, but I'm thinking it's the first.
Sir,
The problem I submitted today Wed 8/03/05 should have read
Find sum of 1/4+2/4+2^2/4+2^3/4 +…+ 2^14/4 …
As I told you r>1 so I thought I’d use Sum =a((1-r^n)/(1-r)).. I used n=14 but now realize I should have used n=15, however, I’m not convinced this is the right approach.
If you factor 1/4 out of the expression you get
1/4 ( 2^0 + 2^1 + 2^2 + ... ).
sum ( r^k, k from 0 to n) is (1 - r^(n+1)) / ( 1 - r). So the above sum would be 1/4 (1 - r^15) / ( 1 - r) = 1/4 ( 1 - 2^15) / (1 - 2) = 1/4 ( 2^15 - 1).
Now, if I look at problem #2… 15 (top) sigma ( bottom) n=1, (2/3)^n . I know to Sum the sequence from 1 through 15 with each having a difference of the ratio (2/3)
sum((2/3)^n, n from 1 to 15) is the same as 2/3 sum((2/3)^n, n from 0 to 14) -- either way you get (2/3) + ( 2/3)^2 + ... + (2/3)^15.
sum((2/3)^n, n from 0 to 14) = ( 1 - (2/3)^15) / ( 1 - 2/3), whichever way the formula is stated. And 2/3 of this is the result you got.
I tried (2/3)((1-(2/3)^15)/(1-(2/3)) assuming that (2/3) was not only my ratio but (a1) as well. This yielded 1.99543 as in the book .
I am lost as to why the equation a((1-r^n)/(1-r)) works in the instance for Prob #2 and not in the first problem above. The only difference I see in the 2nd prob. is that the (a1) term is the same as the ratio, where the 1st problem it has a different 1st term to ratio. If this is the reason then the equation Sum =a((1-r^n)/(1-r)) is conditional is it not !!!
If, in the first Prob I input (1/4) 2*((1-2^14)/(1-2)) it comes out very close to the right answer "off by .25 "units."
this should be (1/4) ((1-2^15)/(1-2)), which comes out correct.
Basically prob.# 1 is solved by Sum = a((1-r^(n+1)/(1-r)) where prob #2 is solved by Sum =a((1-r^n)/(1-r)) where (a) and ratio are equal in prob #2...
Can you clarify my confusion…