Use this tool to solve for any missing variable in a finite Geometric Sequence (Series), such as the first term, common ratio, number of terms, or the total sum of the sequence.
Geometric Sequence Sum Calculator
Geometric Sequence Sum Formula
Variables Explained
- First Term ($A$): The starting value of the sequence (e.g., the initial deposit or the first payout).
- Common Ratio ($R$): The constant factor by which each term is multiplied to get the next term. Often expressed as $(1 + \text{rate of growth})$.
- Number of Terms ($N$): The count of elements in the sequence. Must be a positive integer.
- Sum of N Terms ($S_n$): The total result obtained by adding all $N$ terms of the sequence together.
What is a Geometric Sequence Calculator?
A Geometric Sequence Calculator is a specialized tool used to analyze and solve problems related to sequences where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio ($R$). This type of sequence is fundamental in many areas, particularly finance (compound interest, annuities) and advanced mathematics.
This specific calculator focuses on the sum of a finite geometric series ($S_n$). It allows you to input three known variables ($A, R, N, \text{or } S_n$) and solve for the missing fourth variable. For instance, you can determine the total value of a savings plan (the sum, $S_n$) given the first payment ($A$), the rate of return ($R$), and the number of periods ($N$).
Understanding this calculation is crucial for anyone evaluating long-term financial growth, calculating the total distance traveled by a bouncing ball, or solving certain types of probability problems.
How to Calculate Geometric Sum (Example)
Suppose you invest $1,000 today, and for the next 4 years, you will receive payments that grow by 5% annually. $A = \$1,000$. $R = 1.05$. $N = 5$ (The initial investment + 4 growing payments = 5 total terms).
- Identify Variables: $A=1000$, $R=1.05$, $N=5$. We are solving for $S_n$.
- Calculate $R^N$: $1.05^5 \approx 1.276281$.
- Apply Numerator: $1 – R^N = 1 – 1.276281 = -0.276281$.
- Apply Denominator: $1 – R = 1 – 1.05 = -0.05$.
- Find the Multiplier: $\frac{1 – R^N}{1 – R} = \frac{-0.276281}{-0.05} \approx 5.525628$.
- Calculate the Sum ($S_n$): $S_n = A \times 5.525628 = 1000 \times 5.525628 = \$5,525.63$.
Frequently Asked Questions (FAQ)
Is this the same as an Annuity Calculator?
A geometric sequence is the mathematical foundation for an annuity calculation. While very similar, a pure annuity calculator typically uses time value of money functions (like PV and FV), whereas this tool solves the core mathematical series.
What happens if the Common Ratio ($R$) is 1?
If $R=1$, the formula denominator becomes zero, which is undefined. Mathematically, the sequence is an arithmetic series where all terms are the same ($A, A, A, \dots$). The sum is simply $S_n = A \times N$. Our calculator handles this boundary condition automatically.
Can I solve for the Number of Terms ($N$)?
Solving for $N$ or $R$ requires complex algebraic manipulation involving logarithms or iterative numerical methods (Goal Seeking). To maintain robustness, this calculator primarily calculates the Sum ($S_n$) when $A, R, N$ are given, and performs a consistency check if $S_n$ is also provided.
What is the difference between a Sequence and a Series?
A sequence is the ordered list of numbers (e.g., 2, 4, 8, 16). A series is the sum of the numbers in that sequence (e.g., $2 + 4 + 8 + 16 = 30$). This calculator solves for the sum, hence it calculates a series.