Sum of Total Terms of Geometric Progression Calculator

Geometric Progression Sum Formula:

\[ S_{Total} = \frac{a \times (r^{n_{Total}} - 1)}{r - 1} \]

Sum of Total Terms of Progression (STotal):

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1. What is the Sum of Geometric Progression?

The sum of a geometric progression is the total of all terms in a geometric sequence. A geometric progression is a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

2. How Does the Calculator Work?

The calculator uses the geometric progression sum formula:

\[ S_{Total} = \frac{a \times (r^{n_{Total}} - 1)}{r - 1} \]

Where:

\( a \) — First term of the progression
\( r \) — Common ratio of the progression
\( n_{Total} \) — Number of total terms in the progression

Explanation: This formula calculates the sum of a finite geometric series where the common ratio is not equal to 1.

3. Importance of Geometric Progression Sum

Details: Calculating the sum of geometric progressions is essential in various mathematical applications, including financial calculations, population growth models, computer algorithms, and physics problems involving exponential growth or decay.

4. Using the Calculator

Tips: Enter the first term (a), common ratio (r), and number of terms (nTotal). The common ratio cannot be 1, and the number of terms must be a positive integer.

5. Frequently Asked Questions (FAQ)

Q1: What if the common ratio is 1?
A: If the common ratio is 1, the progression becomes an arithmetic progression with constant terms, and the sum is simply \( a \times n_{Total} \).

Q2: Can this formula handle negative common ratios?
A: Yes, the formula works for both positive and negative common ratios, as long as the ratio is not equal to 1.

Q3: What about infinite geometric series?
A: This calculator is for finite geometric progressions. For infinite series, the sum converges only when |r| < 1, and the formula is \( S = \frac{a}{1 - r} \).

Q4: Are there practical applications of this formula?
A: Yes, geometric progression sums are used in compound interest calculations, computer science algorithms, population studies, and many scientific applications.

Q5: What if the number of terms is very large?
A: The calculator can handle large numbers, but extremely large values may cause computational limitations depending on the system.