Sum of Geometric Sequences

[robo_1337]

Can anyone help me out with this, my teacher hasn't explained it very well and, frankly, I hate the textbooks explanation for things...

thanks,

robo

[mikehepro]

It's applying the equation, what happenes is the next number of the sequence have a ratio to the previous one, R, so if the first term is 3, and there's a ratio of 3, the next number will be 9, the next number after that will be 27. If you want to find the sum of such sequences, say the sum of first 10 terms, you find what a (the first term) is, the r(ratio) and use 10 as n, (the number of terms), then all you have to do is sub these values into the equation, and that's it. Beware that there's an equation for r<1 and there's another one for r>1, they are different.

[Yendall]

Been a while since i've done this, but I believe these are the formulae:

=

=

[plato]

Been a while since i've done this, but I believe these are the formulae:

=

=

The first formula is correct for the sum of n terms of a geometric sequence.

The second formula should read   a/(1-r) rather than 1/(1-r) and is then the sum to infinity of a geometric sequence where 0<r<1.

Edit: Exceptions...
If r>=1, the geometric sequence would be divergent and the infinite sum cannot be found unless the value of a is a fraction.

NOTE:  If r is a fraction, difficulties can arise if a is also negative.
For example, if a=-4 and r=0.5, then the second term would be the square root of -4 which is not possible in real numbers.

If a and r are both integers, then the sequence is divegent and a sum to infinity cannot be found, For example, a=2 and r=-2 would give the sequence 2, 1/4, 16, 1/256 ....

If r = 1, then the sequence is again divergent. For example, a=2 and r=1 gives a sequence 2,2,2,2... which has no sum to infinity.

If r = -1, then we get a sequence such as 2, -2, 2, -2, ........ which can give at least two possible answers for an infinite sum:
Either (2-2) + (2-2) + (2-2) .....  = 0
or       2 - (2-2) - (2-2) - (2-2).... = 2

If r < -1 and .........

All such exceptions to the geometric series formulas given in the study design could not be examined but some may find them worthy of exploration.

[Yendall]

The first formula is correct for the sum of n terms of a geometric sequence.

The second formula should read   a/(1-r) rather than 1/(1-r) and is then the sum to infinity of a geometric sequence where -1<r<0 or 0<r<1.

If r>=1 or r<=-1, the geometric sequence would be divergent and the infinite sum cannot be found.

Ah right, sorry I must've missed something. Thanks for clearing that up for Robo

[robo_1337]

Is there really any difference between the formulas?
The ratio appears to be the same for both = and = .
ie. = =

robo

[Professor Polonsky]

No, it's the same thing.



The latter is probably easier to use when

There's also the sum to infinity, which only applies when