It's not possible to find the common difference of an arithmetic sequence solely from the sum of its terms. The sum, by itself, doesn't provide enough information. You need additional information, such as at least two terms in the sequence, or the first term and the number of terms.
Here's why and how we can approach finding the common difference:
Understanding the Problem
The sum of an arithmetic sequence is determined by the first term, the number of terms, and the common difference. The formula for the sum of an arithmetic sequence is:
- Sₙ = n/2 [2a + (n - 1)d]
Where:
- Sₙ = Sum of the first 'n' terms
- n = Number of terms
- a = First term of the sequence
- d = Common difference
This formula shows that the sum depends on multiple factors, meaning you can't work backward to get 'd' with only 'Sₙ'.
How to Find the Common Difference Using Different Approaches
While the sum alone isn't enough, here are situations where you can find the common difference:
Using Two Consecutive Terms
- According to the reference, the common difference is the value between each successive number in an arithmetic sequence.
- The formula for finding the common difference is: d = a(n) - a(n - 1)
- Where a(n) is the nth term, and a(n - 1) is the previous term.
Example: If we have the sequence 2, 5, 8...
- We can calculate 'd' as 5 - 2 = 3 or 8-5 =3. The common difference is 3.
Using Two Non-Consecutive Terms and their Positions
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Identify Two Terms: Let's say you have the terms aₘ and aₙ where 'm' and 'n' are their respective positions in the sequence and m is not equal to n.
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Find the Difference in Terms: Subtract the smaller term from the larger term: aₙ - aₘ
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Find the Difference in Positions: Subtract the smaller position from the larger position: n - m
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Calculate the Common Difference: Divide the difference in terms by the difference in positions.
* **d = (*a*ₙ - *a*ₘ ) / (n - m)**Example: In an arithmetic sequence, the 4th term is 11 and the 7th term is 20. Calculate the common difference.
- a₇ = 20
- a₄ = 11
- 7 - 4 = 3
- d = (20 - 11) / (7-4)
- d= 9/3 = 3
The common difference is 3.
Using the First Term, nth Term and Number of terms
- Identify the terms: If you know the first term a₁, the nth term aₙ, and the number of terms n in the sequence.
- Use the Formula: We can use the formula aₙ = a₁ + (n-1)d, where aₙ is nth term, a₁ is the first term and d is the common difference. Rearrange the formula to solve for the common difference: d = (aₙ - a₁) / (n - 1)
Example: If the first term of an arithmetic sequence is 2, the 5th term is 14, and n=5. find the common difference.
- a₁= 2
- a₅ = 14
- n=5
- d = (14-2) / (5-1)
- d = 12 /4
- d= 3
The common difference is 3.
Conclusion
In conclusion, you cannot directly calculate the common difference from the sum of an arithmetic series alone. You'll need at least two terms or information about the first term, last term and the number of terms. The formula d = a(n) - a(n - 1) is the most direct method when you have two consecutive terms or you can use the formula d= (aₙ - a₁) / (n - 1) if you know the first term, last term, and the number of terms.
