Finding the sum of a geometric series

In summary, the conversation discusses finding the ratio, r, in a geometric series formula. The ratio is the item that is raised to the power of the sequence index number. The general term is indexed by i and can be used to find the ratio between consecutive terms. The formula for finding the sum of a geometric series is first term*(1-r^n)/(1-r). Wolfram Mathmatica provides a helpful explanation for geometric series.
  • #1
umzung
21
0
Homework Statement
Calculate:
Relevant Equations
The sum of 5 x 10^i, from i=0 to i=n-1
I'm using the sum of a geometric series formula, but I'm not sure how to find the ratio, r. The n is confusing me.

The solution is below, but I'm having trouble with the penultimate step.

Screenshot 2019-10-20 11.31.30.png
 
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  • #2
The ratio is 10. In a geometric seqence the ratio of consecutive terms is always the item that is raised to the power of the sequence index number (j in this case).
 
  • #3
Here r is the ratio of consecutive terms. The general term is 5x10^i ; it is indexed by i. Can you use this to find the ratio between consecutive terms?
 
  • #4
Consider this:
##\sum_{i=0}^{n-1} ab^i = \sum_{i=0}^\infty ab^i - \sum_{i=n}^\infty ab^i##
 
  • #5
andrewkirk said:
The ratio is 10. In a geometric seqence the ratio of consecutive terms is always the item that is raised to the power of the sequence index number (j in this case).
Thanks. The formula is first term*(1-r^n)/(1-r). How does 1-r become reversed in the solution?
 
  • #6
Try multiplying the top and bottom of the expression by -1.
 
  • #7
Wolfram

Wolfram Mathmatica gives a nice explanation of geometric series.
 

Related to Finding the sum of a geometric series

1. What is a geometric series?

A geometric series is a sequence of numbers where each term is obtained by multiplying the previous term by a constant factor. For example, the series 1, 2, 4, 8, 16, 32... is a geometric series with a common ratio of 2.

2. How do you find the sum of a geometric series?

To find the sum of a geometric series, you can use the formula S = a(1-r^n)/(1-r), where S is the sum, a is the first term, r is the common ratio, and n is the number of terms in the series. Alternatively, you can also use the formula S = a/(1-r), where S is the sum, a is the first term, and r is the common ratio, if the number of terms goes to infinity.

3. What is the difference between a finite and infinite geometric series?

A finite geometric series has a fixed number of terms, while an infinite geometric series continues on forever. In a finite series, the sum can be calculated using the formula S = a(1-r^n)/(1-r) where n is the number of terms. In an infinite series, the sum can be calculated using the formula S = a/(1-r), where a is the first term and r is the common ratio.

4. Can a geometric series have a negative common ratio?

Yes, a geometric series can have a negative common ratio. This means that the series will alternate between positive and negative numbers. For example, the series 1, -2, 4, -8, 16... is a geometric series with a common ratio of -2.

5. How can I use a geometric series in real-life situations?

Geometric series can be used to model real-life situations such as population growth, compound interest, and depreciation. For example, a population that doubles every year can be represented by a geometric series with a common ratio of 2. This can also be applied to investments with compound interest, where the principal amount grows exponentially over time.

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