Finding the Common Difference in an Arithmetic Progression
In mathematics, an arithmetic progression (A.P.) is a sequence of numbers such that the difference between any two successive members is a constant. This constant is known as the common difference. In this article, we will explore the process of finding the common difference when given the first and last terms and the sum of all terms in an arithmetic progression. We will use a specific example to illustrate the steps involved.
Understanding the Arithmetic Progression (A.P.)
An arithmetic progression is defined by three key elements: the first term (a), the common difference (d), and the last term (l). The sum (S) of the terms in an A.P. can be calculated using the formula:
Formula: Sum n/2 × (first term last term)
Given Data
In our problem, the first term a is given as 7, the last term l is given as 49, and the sum of all terms S is 400. Using the sum formula, we can find the number of terms (n) in the arithmetic progression.
Step 1: Calculate the Number of Terms (n)
The sum formula for an arithmetic progression is:
Sum n/2 first term last term
Given:
Sum 400, first term a 7, last term l 49
Substitute the given values into the sum formula:
400 n/2 7 49
Simplify the expression inside the parentheses:
400 n/2 56
Divide both sides of the equation by 56:
400/56 n/2
Simplify the fraction:
100/14 n/2
Further simplification:
100/14 n/2 → 100/14×2 n
Simplify:
n 100/7
Therefore, the number of terms in the arithmetic progression is 100/7.
Step 2: Calculate the Common Difference (d)
The formula to find the common difference is:
d (last term - first term) / (n - 1)
Substitute the given values into the formula:
d (49 - 7) / (100/7 - 1)
Simplify the numerator and denominator:
d 42 / (100/7 - 1)
Convert 1 to a fraction with the same denominator:
d 42 / (100/7 - 7/7)
Perform the subtraction in the denominator:
d 42 / (93/7)
Therefore, the common difference of the arithmetic progression is 42.
Conclusion
By following these steps, you can determine the common difference in an arithmetic progression when given the first and last terms and the sum of all terms. This example demonstrates how to use the sum formula to calculate the number of terms and the common difference.
Key Takeaways
Arithmetic progression is a sequence of numbers with a constant difference. The formula to find the common difference is d (last term - first term) / (n - 1), where n is the number of terms. The sum of terms in an arithmetic progression can be calculated using the formula Sum n/2 × (first term last term).Further Reading
For more information on arithmetic progressions and related mathematical concepts, explore our articles on:
Arithmetic Progression Series Sum of Terms in an A.P. Understanding the Common Difference