Find The Sum Of The Finite Geometric Sequence Calculator

Easily calculate the sum of the first 'n' terms of a geometric sequence using our sum of finite geometric sequence calculator. Input the first term (a), common ratio (r), and number of terms (n) to get the sum (S_n).

What is the Sum of a Finite Geometric Sequence?

The sum of a finite geometric sequence (also known as the sum of a finite geometric series) is the total obtained by adding up the first 'n' terms of a geometric sequence. A geometric sequence 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, if the first term (a) is 2 and the common ratio (r) is 3, the sequence starts: 2, 6, 18, 54, 162… The sum of the first 5 terms would be 2 + 6 + 18 + 54 + 162 = 242. Our sum of finite geometric sequence calculator helps you find this sum quickly.

This concept is useful in various fields, including finance (for calculating compound interest or annuities), physics, computer science, and biology, where quantities grow or decay exponentially over discrete steps.

Common misconceptions include confusing it with an arithmetic sequence (where terms are added by a common difference) or an infinite geometric series (which may or may not have a finite sum).

Sum of Finite Geometric Sequence Formula and Mathematical Explanation

The formula to find the sum of the first n terms of a finite geometric sequence (S_n) is:

S_n = a(1 – rⁿ) / (1 – r)

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

This formula is valid when the common ratio r is not equal to 1. If r = 1, then each term is equal to 'a', and the sum is simply S_n = n * a.

Derivation:

Let S_n = a + ar + ar² + … + ar^n-1.
Multiply by r: rS_n = ar + ar² + ar³ + … + arⁿ.
Subtract rS_n from S_n:
S_n – rS_n = (a + ar + … + ar^n-1) – (ar + ar² + … + arⁿ)
S_n(1 – r) = a – arⁿ
S_n(1 – r) = a(1 – rⁿ)
So, S_n = a(1 – rⁿ) / (1 – r) (for r ≠ 1).

Variable	Meaning	Unit	Typical Range
Sn	Sum of the first n terms	Depends on 'a'	Calculated
a	First term	Any unit	Any real number
r	Common ratio	Dimensionless	Any real number (formula differs if r=1)
n	Number of terms	Dimensionless	Positive integer

Variables used in the sum of finite geometric sequence formula.

Practical Examples (Real-World Use Cases)

Example 1: Savings Growth

Suppose you save $100 in the first month and decide to increase your savings by 5% each month for 12 months. This is a geometric sequence with a = 100, r = 1.05, and n = 12. To find the total amount saved after 12 months, we calculate the sum of this finite geometric sequence:

S₁₂ = 100 * (1 – 1.05¹²) / (1 – 1.05) ≈ 100 * (1 – 1.795856) / (-0.05) ≈ 100 * (-0.795856) / (-0.05) ≈ $1591.71

You would have saved approximately $1591.71 after 12 months.

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 before it comes to rest (theoretically, after a very large number of bounces, but let's consider the first 8 bounces for a finite sum)?

The distances are: Down 10, Up 7, Down 7, Up 4.9, Down 4.9… Total distance = 10 + 2*(7 + 4.9 + 3.43 + … for 7 terms after the first drop). The sum of the upward (or downward after first drop) distances is a geometric series with a=7, r=0.7, n=7. S₇ = 7 * (1 – 0.7⁷) / (1 – 0.7) ≈ 7 * (1 – 0.0823543) / 0.3 ≈ 7 * 0.9176457 / 0.3 ≈ 21.41 meters. Total distance ≈ 10 + 2 * 21.41 = 10 + 42.82 = 52.82 meters after 8 bounces (1 drop + 7 up/down cycles).

How to Use This Sum of Finite Geometric Sequence Calculator

1. Enter the First Term (a): Input the initial value of your sequence.
2. Enter the Common Ratio (r): Input the factor by which each term is multiplied to get the next. Ensure r is not 1 (the calculator handles r=1, but the main formula shown is for r≠1).
3. Enter the Number of Terms (n): Input how many terms of the sequence you want to sum. This must be a positive integer.
4. View the Results: The calculator automatically updates and displays the sum (S_n), intermediate values like rⁿ, and the formula used.
5. Examine the Table and Chart: The table shows each term and the running total, while the chart visually represents the growth or decay of terms and the sum.

The results help you understand how the sum accumulates and the magnitude of individual terms in the finite geometric sequence.

Key Factors That Affect Sum of Finite Geometric Sequence Results

• First Term (a): The larger the initial term, the larger the sum, proportionally.
• Common Ratio (r):
  • If |r| > 1, the terms grow exponentially, and the sum can become very large quickly as n increases.
  • If |r| < 1, the terms decrease, and the sum approaches a limit as n increases (related to the sum of an infinite geometric series).
  • If r is positive, all terms have the same sign as 'a'.
  • If r is negative, the terms alternate in sign.
  • If r = 1, the sum is simply n*a.
  • If r = -1, the sum alternates between 'a' and 0 (for n even/odd).
• Number of Terms (n): The more terms you sum, the larger the absolute value of the sum will generally be, especially if |r| > 1. If |r| < 1, the sum will approach a finite limit.
• Sign of 'a' and 'r': The signs of 'a' and 'r' determine the signs of the terms and thus the direction of the sum's growth or oscillation.

Frequently Asked Questions (FAQ)

Q: What happens if the common ratio (r) is 1? A: If r = 1, each term is equal to 'a'. The sum of the first 'n' terms is simply n * a. Our sum of finite geometric sequence calculator handles this case.

Q: What if the common ratio (r) is negative? A: The terms of the sequence will alternate in sign (e.g., a, -ar, ar², -ar³,…). The formula S_n = a(1 – rⁿ) / (1 – r) still applies.

Q: Can the number of terms (n) be zero or negative? A: No, the number of terms 'n' must be a positive integer, as it represents the count of terms you are summing.

Q: How does this relate to the sum of an infinite geometric series? A: If the absolute value of the common ratio |r| < 1, as 'n' becomes very large (approaches infinity), rⁿ approaches 0. The sum of the infinite geometric series then becomes S = a / (1 – r). Our calculator is for a finite number of terms.

Q: Can I use this calculator for financial calculations like annuities? A: Yes, the formula for the present or future value of an ordinary annuity is derived from the sum of a finite geometric sequence, where 'a' is the first payment/deposit adjusted for interest, and 'r' involves the interest rate. See our annuity calculator for more specific financial calculations.

Q: What if 'a' is zero? A: If the first term 'a' is 0, then all terms are 0, and the sum will always be 0.

Q: How accurate is this sum of finite geometric sequence calculator? A: The calculator uses standard floating-point arithmetic, so it's very accurate for most practical purposes. Extremely large 'n' or 'r' values might lead to precision limitations inherent in computer calculations.

Q: Where is the sum of a finite geometric sequence used? A: It's used in finance (compound interest, annuities), physics (decay processes), computer science (algorithms), biology (population growth models), and more.

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