Question

What is the relevance of an infinite sum when talking about geometric and arithmetic sequences? (ex. 76 infinity)
What and how do you use it?

Answer 1

If the sequences that is generated approaches zero, then there is a possibility that the sum of the sequence will even out to some solid value ...

a1 + a2 + a3 + .... + 0, this is a notion called convergence.

Answer 2

if the sequence just keep getting bigger then it diverges:

a1 + a2 + a3 + ... + inf = inf

Answer 3

But what if a problem says
a geometric sequence u1, u2, u3 ... has u1 = 27 and a sum to infinity os 81/2

what happens?

Answer 4

well we can generate the infinite formula ...

would you agree that the sum of a geometric sequence with a seed of a, and a finite number of terms k

a + ar + ar^2 + ar^3 + ar^4 + ... + ar^(k-1)

??

Answer 5

isnt it
a1/1-r

Answer 6

But what's r?

Answer 7

well, thats the formula i was going to generate .. so yeah

Answer 8

do the algebra:\[\frac{81}{2}=\frac{27}{1-r}\]
colve for r

Answer 9

to me it looks like:
\[r=1-\frac{27}{81/2}\]

Answer 10

what does r stand for? like whats the relevance?

Answer 11

i used it in the last question ... do you remember what i called it and what i used it for?

Answer 12

since

a1 r = a2 ; r = a2/a1
a2 r = a3 ; r = a3/a2
a3 r = a4 ; r = a4/a3

since r is held constant, then there must be a common ratio, r, that relates each new term with the last term used.

Answer 13

r is called the common ratio and is defined to be \[\large r=\frac{a_{n+1}}{a_n}\]