Recursion Formula Calculator

The Recursion Formula Calculator allows users to compute the nth term value, sum, and convergence of both geometric and arithmetic sequences based on user-defined initial values, common ratios, and recursion types.

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How to Use the Recursion Formula Calculator

This guide will help you navigate and utilize the Recursion Formula Calculator effectively. Follow these steps to input your data, choose the formula type, and interpret the results.

Step 1: Input Initial Information

  1. Initial Value (a₁):

    Locate the field labeled “Initial Value (a₁)”. Enter a numerical value that represents the first term of your sequence. Ensure this value is between -1,000,000 and 1,000,000 as per the validation criteria.

  2. Common Ratio (r):

    Next, find the field labeled “Common Ratio (r)”. Input the common ratio for your sequence, which should be a number between -100 and 100. You can specify the common ratio to two decimal places using steps of 0.01.

  3. Calculate up to nth Term:

    Enter the term position (n) you wish to calculate up to in the field provided. This number should be an integer between 1 and 20.

  4. Recursion Type:

    Select the type of sequence you are dealing with. The options available are:

    • Geometric Sequence (aₙ = a₁ × rⁿ⁻¹)
    • Arithmetic Sequence (aₙ = a₁ + (n-1)×r)

    Use the dropdown menu to choose the appropriate type for your calculation.

Step 2: Calculate and Review Results

  1. nth Term Value:

    Based on your entries, the calculator will compute the value of the nth term in the sequence. The formula used varies depending on whether it’s a geometric or an arithmetic sequence. The result will be displayed with up to four decimal places.

  2. Sum of First n Terms:

    This field shows the sum of the first n terms of your sequence, calculated using the appropriate summation formula for geometric or arithmetic sequences.

  3. Convergence Value (if |r| < 1):

    For geometric sequences, if the absolute value of the common ratio is less than 1, a convergence value is calculated and presented. If the sequence does not meet this criterion or is not geometric, it will indicate that it does not converge.

By following these steps, you will be able to effectively use the Recursion Formula Calculator to determine key aspects of geometric and arithmetic sequences.