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Geometric Series Calculator.

Calculates the sum of a finite geometric series given the first term, common ratio, and number of terms.

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Your inputs

How it works

  1. 1

    Identify the first term (a), common ratio (r), and number of terms (n).

  2. 2

    Check if r equals 1; if so, the sum is a * n.

  3. 3

    Use the formula S = a * (1 - r^n) / (1 - r).

  4. 4

    Compute r^n, then multiply by a and divide by (1 - r).

first_term * (1 - pow(common_ratio, num_terms)) / (1 - common_ratio)

Frequently asked questions

What if the common ratio is 1?

The formula has a division by zero. In that case, the sum is simply the first term multiplied by the number of terms.

Can the common ratio be negative?

Yes, negative ratios cause the terms to alternate in sign. The formula still works as long as r is not 1.

What is the difference between finite and infinite geometric series?

A finite series has a fixed number of terms. An infinite series converges only if |r| < 1, and its sum is a / (1 - r).

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Sum of series

31

Last term16.00
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How it works

Calculates the sum of a finite geometric series given the first term, common ratio, and number of terms.

  1. Identify the first term (a), common ratio (r), and number of terms (n).
  2. Check if r equals 1; if so, the sum is a * n.
  3. Use the formula S = a * (1 - r^n) / (1 - r).
  4. Compute r^n, then multiply by a and divide by (1 - r).

Formulas

The math behind this calculator, written out so you can verify the result.

Finite geometric series sum

S = a * (1 - r^n) / (1 - r)

This gives the sum of the first n terms of a geometric sequence.

Example:

Input: a = 1, r = 2, n = 5

Calculation: 1 * (1 - 2^5) / (1 - 2) = 1 * (1 - 32) / (-1) = 31

Result: 31

Infinite geometric series sum (|r| < 1)

S = a / (1 - r)

When the common ratio is between -1 and 1, the sum of infinitely many terms approaches this value.

Example:

Input: a = 1, r = 0.5

Calculation: 1 / (1 - 0.5) = 2

Result: 2

Real-world use cases

Where this calculation shows up in everyday life.

Compound interest

The total value of an annuity or a series of payments can be modeled as a geometric series.

Example: Calculating the future value of regular deposits.

Fractal geometry

Areas or lengths in fractals often follow geometric series.

Example: Computing the total area of a Sierpinski triangle.

Economics and finance

Multiplier effects and present value calculations use geometric series.

Example: Estimating the total spending from an initial investment.

Tips and common mistakes

Tips

  • If the common ratio is close to 1, the sum grows quickly; double-check your inputs.
  • For negative ratios, the sum can oscillate; use parentheses carefully.
  • Use the last term result to verify the series progression.
  • Remember that the formula only works when r ≠ 1.

Common Mistakes to Avoid

  • Forgetting to handle r = 1 separately, leading to division by zero.
  • Using the infinite series formula when |r| ≥ 1, which diverges.
  • Miscounting the number of terms; n is the count of terms, not the exponent of the last term.

Assumptions and limitations

  • Use the stated inputs and units.
  • Results are estimates for planning and education.
  • Check measurements and source data before making an important decision.