Nth Term of Arithmetic Progression given Sum of First N Terms Calculator

Formula Used:

\[ T_n = \frac{2 \times S_n}{n} - a \]

Sum of First N Terms of Progression (S_n):

Index N of Progression (n):

First Term of Progression (a):

Nth Term of Progression (T_n):

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1. What is the Nth Term of Arithmetic Progression Formula?

The formula \( T_n = \frac{2 \times S_n}{n} - a \) calculates the nth term of an arithmetic progression when given the sum of the first n terms and the first term of the progression. This formula is derived from the standard arithmetic progression properties.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ T_n = \frac{2 \times S_n}{n} - a \]

Where:

\( T_n \) — Nth Term of Progression
\( S_n \) — Sum of First N Terms of Progression
\( n \) — Index N of Progression
\( a \) — First Term of Progression

Explanation: This formula allows you to find any term in an arithmetic progression when you know the sum of the first n terms and the first term of the sequence.

3. Importance of Nth Term Calculation

Details: Calculating the nth term of an arithmetic progression is essential in various mathematical applications, including sequence analysis, pattern recognition, and solving problems involving arithmetic sequences in algebra and number theory.

4. Using the Calculator

Tips: Enter the sum of the first n terms, the index n, and the first term of the progression. All values must be valid (n must be a positive integer greater than 0).

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.

Q2: When is this formula particularly useful?
A: This formula is useful when you know the sum of the first n terms and need to find a specific term in the sequence without knowing the common difference.

Q3: Can this formula be used for geometric progressions?
A: No, this formula is specific to arithmetic progressions. Geometric progressions have different formulas for calculating terms and sums.

Q4: What if n is not a positive integer?
A: The index n must be a positive integer as it represents the position of a term in the sequence.

Q5: How accurate are the results from this calculator?
A: The calculator provides results with up to 6 decimal places precision, making it suitable for most mathematical applications.