S R 1 I N-1 I Calculator

This calculator computes the sum of a series using the formula S = r(1 - i^n)/(1 - i). It's commonly used in physics, engineering, and financial calculations where periodic payments or values need to be summed.

What is S = r(1 - i^n)/(1 - i)?

The formula S = r(1 - i^n)/(1 - i) represents the sum of a geometric series where:

S is the total sum
r is the first term of the series
i is the common ratio between terms
n is the number of terms

This formula is derived from the geometric series summation formula and is particularly useful when calculating the present value of an annuity, future value of an investment with periodic contributions, or any scenario involving a series of equal payments or values.

Key Assumptions

The formula assumes that the series is finite and that the common ratio i is not equal to 1. If i = 1, the series becomes arithmetic and requires a different calculation method.

How to Use This Calculator

Using the calculator is straightforward:

Enter the first term (r) of your series
Enter the common ratio (i) between terms
Enter the number of terms (n) in the series
Click "Calculate" to compute the sum
Review the result and chart visualization

The calculator will display the total sum and provide a visual representation of how the series grows with each term.

The Formula Explained

The formula S = r(1 - i^n)/(1 - i) can be broken down as follows:

Formula Breakdown

The numerator (1 - i^n) represents the difference between the first term and the nth term. The denominator (1 - i) represents the common ratio. When divided, this gives the sum of all terms in the series.

This formula is particularly useful in financial calculations where payments are made at regular intervals, such as monthly contributions to a savings account or regular payments on a loan.

Worked Examples

Example 1: Simple Geometric Series

Suppose you have a series with first term r = 2, common ratio i = 0.5, and n = 4 terms. The sum would be calculated as:

Calculation

S = 2(1 - 0.5^4)/(1 - 0.5) = 2(1 - 0.0625)/0.5 = 2(0.9375)/0.5 = 3.75

The total sum of this series is 3.75.

Example 2: Financial Application

In finance, this formula can be used to calculate the present value of an annuity. If you expect to receive $1000 at the end of each year for 5 years with a discount rate of 5%, the present value would be calculated using a similar approach.

Practical Applications

The S = r(1 - i^n)/(1 - i) formula has several practical applications:

Calculating the present value of an annuity
Determining the future value of an investment with periodic contributions
Analyzing the growth of a series with a constant ratio between terms
Modeling scenarios where values increase or decrease by a fixed percentage

Understanding this formula can help in financial planning, engineering calculations, and scientific research involving series and sequences.

Frequently Asked Questions

What happens if the common ratio i is 1?

If the common ratio i is 1, the series becomes arithmetic, and the formula S = r(1 - i^n)/(1 - i) will not work because the denominator becomes zero. In this case, you should use the arithmetic series formula S = n*r/2.

Can this formula be used for infinite series?

No, this formula is specifically for finite series. For infinite series where the common ratio |i| < 1, you would use the formula S = r/(1 - i).

What units should be used for the inputs?

The units for r and the resulting sum S will depend on the context of your calculation. For financial applications, r might represent dollars, and S would also be in dollars. For other applications, ensure all inputs use consistent units.