To find the common difference of an arithmetic progression when the sum of its terms is given, we can use a specific formula derived from the sum formula. Here's a step-by-step guide incorporating the information from the reference:
The reference provides a method to calculate the common difference (d) when the sum of n terms (Sn) and the first term (a1) are known. The steps are as follows:
Steps to Calculate Common Difference:
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Start with the sum formula and modify it: We know the sum of n terms of an arithmetic progression is given by: Sn = n/2 [2a1 + (n-1)d]
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Isolate the terms involving 'd': The steps to calculate the common difference (d) as per the reference are:
- Step 1: Sn × 2/n = 2a1 + (n-1)d (This is derived by multiplying both sides of the standard sum formula by 2 and dividing by n.)
- Step 2: Sn × 2/n - 2a1 = (n-1)d (Subtracting 2a1 from both sides to isolate the term containing d.)
- Step 3: d = (Sn × 2/n - 2a1)/(n-1) or d = (2Sn - 2na1) / (n(n-1)) (Finally, divide by (n-1) to solve for d.)
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Apply the formula: Use the final formula d = (2Sn - 2na1) / (n(n-1)) to calculate the common difference.
Explanation of the Formula:
- Sn represents the sum of the first n terms of the arithmetic progression.
- n represents the number of terms.
- a1 represents the first term of the progression.
- d represents the common difference we are solving for.
Example:
Let's say the sum of the first 5 terms of an arithmetic progression is 40 (Sn = 40, n = 5) and the first term is 2 (a1 = 2). To find the common difference (d):
- d = (2 40 - 2 5 2) / (5 (5 - 1))
- d = (80 - 20) / (5 4)*
- d = 60 / 20
- d = 3
Therefore, the common difference for this arithmetic progression is 3.
Summary:
The process to calculate the common difference is straightforward:
- Know the sum of terms (Sn), the number of terms (n), and the first term (a1).
- Apply the formula d = (2Sn - 2na1) / (n(n-1)).
- Solve for d.
