Find The Sum Of The Finite Geometric Series Calculator

Easily calculate the sum of the first 'n' terms of a geometric sequence using our {primary_keyword}.

Calculator

Term (k)	Term Value (a*r^(k-1))	Cumulative Sum (S_k)
Table will populate after calculation.

Table showing term values and cumulative sums for the geometric series.

What is a {primary_keyword}?

A {primary_keyword} is a tool used to determine the sum (S_n) of the first 'n' terms of a geometric series (also known as a geometric progression or GP). A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r).

For example, the sequence 2, 6, 18, 54, 162 is a geometric series with a first term (a) of 2 and a common ratio (r) of 3. Our calculator helps find the sum of these terms without manually adding them up, which is especially useful for a large number of terms.

This calculator is beneficial for students learning about sequences and series, finance professionals dealing with annuities or compound interest scenarios (which are based on geometric progressions), and anyone needing to sum a finite number of terms growing or decaying at a constant rate.

A common misconception is that the formula works directly for a common ratio (r) of 1. However, when r=1, the series becomes an arithmetic series with a common difference of 0 (e.g., a, a, a, …), and the sum is simply n*a. Our {primary_keyword} handles this case correctly.

{primary_keyword} Formula and Mathematical Explanation

The sum of the first n terms of a finite geometric series is given by the formula:

S_n = a(1 – rⁿ) / (1 – r)     (when r ≠ 1)

Where:

• S_n is the sum of the first n terms
• a is the first term
• r is the common ratio
• n is the number of terms

If the common ratio r = 1, the series is simply a + a + a + … + a (n times), so the sum is:

S_n = n * a     (when r = 1)

Derivation (r ≠ 1):

1. The series is: S_n = a + ar + ar² + … + ar^n-1
2. Multiply by r: rS_n = ar + ar² + ar³ + … + arⁿ
3. Subtract the second equation from the first: S_n – rS_n = a – arⁿ
4. Factor: S_n(1 – r) = a(1 – rⁿ)
5. Divide by (1 – r): S_n = a(1 – rⁿ) / (1 – r)

Variables in the Formula

Variable	Meaning	Unit	Typical Range
a	First term	Varies (unitless, money, etc.)	Any real number
r	Common ratio	Unitless	Any real number
n	Number of terms	Unitless (count)	Positive integers (≥ 1)
Sn	Sum of the first n terms	Same as 'a'	Varies

Practical Examples (Real-World Use Cases)

The concept of a {primary_keyword} applies to various real-world situations, especially in finance and growth modeling.

Example 1: Savings Plan

Suppose you deposit $1000 at the beginning of each year for 5 years into an account that grows at 5% per year due to some fixed rate applied yearly to the deposited amounts, but let's model this as a geometric series for simplicity of contribution growth idea (though it's more like an annuity). If your contributions themselves were designed to grow by 5% each year (you contribute $1000, then $1050, etc.), what's the sum of your contributions over 5 years? Here, a = 1000, r = 1.05 (1+0.05), n = 5. Using the {primary_keyword}, the sum of contributions is S₅ = 1000 * (1 – 1.05⁵) / (1 – 1.05) ≈ $5525.63. (This is the sum of contributions, not the final value with interest on each). A more direct application is looking at the present or future value of an annuity using a annuity calculator.

Example 2: Bouncing Ball

A ball is dropped from a height of 10 meters. Each time it bounces, it reaches 70% of its previous height. What is the total vertical distance traveled by the ball after it hits the ground for the 5th time (considering only the downward + upward distances after the initial drop)? The distances are: 10 (initial drop), then up 7, down 7, up 4.9, down 4.9, etc. The upward and downward distances after the first drop form a geometric series. Upward: 7, 4.9, 3.43, 2.401, 1.6807. Downward is the same. Series for upward/downward: a=7, r=0.7, n=5. Sum = 7 * (1 – 0.7⁵) / (1 – 0.7) ≈ 19.397 m. Total upward = 19.397, total downward after first drop = 19.397. Total distance = 10 + 19.397 + 19.397 = 48.794 meters after 5 bounces and the subsequent upward travel. Our {primary_keyword} can find the sum 7 + 4.9 + … quickly.

How to Use This {primary_keyword} Calculator

1. Enter the First Term (a): Input the initial value of your geometric series.
2. Enter the Common Ratio (r): Input the constant multiplier between terms. If r=1, the calculator will use S_n = n*a.
3. Enter the Number of Terms (n): Input the total count of terms you want to sum, which must be a positive integer.
4. Calculate: The calculator automatically updates as you type, or you can click the "Calculate Sum" button.
5. View Results: The "Sum (S_n)" will be displayed prominently, along with intermediate values like rⁿ, 1-rⁿ, and 1-r, and the formula used.
6. See Table and Chart: The table and chart below the results show the individual term values and how the sum accumulates.
7. Copy Results: Use the "Copy Results" button to copy the input values and results.

The results help you understand not just the final sum, but also how the series grows or shrinks term by term. For financial applications, understanding the rate of growth (r) and the duration (n) is crucial. A compound interest calculator uses similar principles.

Key Factors That Affect {primary_keyword} Results

• First Term (a): The starting value directly scales the sum. A larger 'a' means a larger sum, proportionally.
• Common Ratio (r): This is the most critical factor.
  • If |r| > 1, the terms grow in magnitude, and the sum can become very large quickly as n increases.
  • If |r| < 1, the terms shrink in magnitude, and the sum approaches a finite limit as n increases (see infinite geometric series calculator).
  • If r = 1, the sum is simply n*a.
  • If r is negative, the terms alternate in sign.
• Number of Terms (n): The more terms you sum, the larger the absolute value of the sum will generally be, especially if |r| > 1.
• Sign of 'a' and 'r': The signs of 'a' and 'r' determine the sign of individual terms and the overall sum.
• Proximity of 'r' to 1: When 'r' is close to 1 (but not 1), the denominator (1-r) is small, which can lead to a large sum even for moderate 'a' and 'n'.
• Magnitude of 'n' when |r| < 1: If |r| < 1, rⁿ becomes very small for large 'n', and the sum S_n approaches a/(1-r).

Understanding these factors is key when using the {primary_keyword} for projections or analysis.

Frequently Asked Questions (FAQ)

1. What is a geometric series?

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. Example: 3, 6, 12, 24 (r=2).

2. How is the common ratio (r) calculated if I have the terms?

Divide any term by its preceding term. For example, in 3, 6, 12, r = 6/3 = 2 or r = 12/6 = 2.

3. What happens if the common ratio (r) is 1?

If r=1, the series becomes a, a, a, …, and the sum of n terms is simply n * a. Our {primary_keyword} handles this case.

4. What if the common ratio (r) is negative?

The terms will alternate in sign (e.g., 2, -4, 8, -16). The formula still applies, and our {primary_keyword} calculates the sum correctly.

5. Can the number of terms (n) be a fraction or negative?

No, the number of terms (n) must be a positive integer as it represents a count of terms in the sequence.

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

A finite geometric series has a specific number of terms (n), and we calculate its sum S_n. An infinite geometric series continues forever. It only has a finite sum if |r| < 1, given by S = a / (1 - r). We have an infinite geometric series calculator for that.

7. Where is the sum of a finite geometric series used?

It's used in finance (calculating the future or present value of certain types of annuities), physics (e.g., bouncing ball distance), and computer science (e.g., analyzing algorithms).

8. Can 'a' or 'r' be zero?

If 'a' is zero, all terms are zero, and the sum is zero. The common ratio 'r' is defined as non-zero for a geometric series.