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jdinatale
- 155
- 0
I would like to find a nice formula for [itex]\sum_{k=0}^{n - 1}ar^{4k}[/itex]. I know that [itex]\sum_{k=0}^{n - 1}ar^{k} = a\frac{1 - r^n}{1 - r}[/itex] and was wondering if there was some sort of analogue.
micromass said:Use the formula with [itex]r^4[/itex] instead of r.
A finite geometric series is a sum of numbers where each term is obtained by multiplying the previous term by a fixed number called the common ratio. It has a fixed number of terms and can be written in the form of a1, a1r, a1r2, ..., a1rn-1, where a1 is the first term and r is the common ratio.
The formula for the sum of a finite geometric series is Sn = a1(1-rn)/ (1-r), where Sn is the sum of the first n terms, a1 is the first term, and r is the common ratio. Simply plug in the values and solve for Sn to find the sum.
Yes, a finite geometric series can have a negative common ratio. In this case, the series will alternate between positive and negative numbers, and the sum will depend on the value of the common ratio.
If the common ratio is equal to 1, then the series becomes a simple arithmetic sequence and the sum can be found using the formula Sn = n(a1 + an)/2, where n is the number of terms, and a1 and an are the first and last terms, respectively.
Yes, finite geometric series are commonly used in finance, such as calculating compound interest on loans and investments. They are also used in population growth models and in calculating the total distance traveled in a journey with a constant speed and changing direction.