First Term of Arithmetic Progression Calculator

Formula Used:

\[ a = T_n - ((n - 1) \times d) \]

First Term of Progression (a):

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1. What is the First Term of Arithmetic Progression?

The First Term of an Arithmetic Progression is the initial term from which the sequence begins. It serves as the starting point for the entire progression, with each subsequent term obtained by adding a constant value called the common difference.

2. How Does the Calculator Work?

The calculator uses the arithmetic progression formula:

\[ a = T_n - ((n - 1) \times d) \]

Where:

\( a \) — First Term of Progression
\( T_n \) — Nth Term of Progression
\( n \) — Index N of Progression
\( d \) — Common Difference of Progression

Explanation: This formula calculates the first term by subtracting the accumulated differences from the nth term back to the starting point of the sequence.

3. Importance of First Term Calculation

Details: Determining the first term is essential for understanding the complete arithmetic sequence, predicting future terms, and analyzing patterns in mathematical and real-world progressions.

4. Using the Calculator

Tips: Enter the nth term value, the position index n (must be ≥1), and the common difference. All values must be valid numerical inputs.

5. Frequently Asked Questions (FAQ)

Q1: What is an arithmetic progression?
A: An arithmetic progression is a sequence of numbers where the difference between consecutive terms is constant throughout the sequence.

Q2: Can the first term be negative?
A: Yes, the first term can be any real number - positive, negative, or zero, depending on the progression.

Q3: What if the common difference is zero?
A: If the common difference is zero, all terms in the progression are equal to the first term, creating a constant sequence.

Q4: How is this formula derived?
A: The formula is derived from the general nth term formula of an arithmetic progression: \( T_n = a + (n - 1) \times d \), rearranged to solve for the first term a.

Q5: What are some real-world applications of arithmetic progressions?
A: Arithmetic progressions are used in financial calculations (installments, depreciation), physics (uniform motion), computer science (algorithm analysis), and many other fields.