To find the sum of the common differences of two arithmetic progressions, you first need to determine the common difference of each progression individually, and then add those common differences together.
Understanding Arithmetic Progressions and Common Difference
An arithmetic progression is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference, often denoted as 'd'.
How to Find the Common Difference
- Identify two consecutive terms: Choose any two terms that are next to each other in the progression.
- Subtract the earlier term from the later term: The result is the common difference. So, d = (Later Term) - (Earlier Term).
Calculating the Sum of Common Differences
Once you have the common difference for both progressions, summing them is straightforward.
Steps:
- Find the common difference (d1) of the first arithmetic progression.
- Find the common difference (d2) of the second arithmetic progression.
- Add the two common differences together: d1 + d2. This is the sum you're looking for.
Example:
Let's say you have two arithmetic progressions:
- Progression 1: 2, 4, 6, 8, ...
- Progression 2: 1, 5, 9, 13, ...
- Common difference of Progression 1 (d1): 4 - 2 = 2
- Common difference of Progression 2 (d2): 5 - 1 = 4
- Sum of the common differences: 2 + 4 = 6
Therefore, the sum of the common differences of these two progressions is 6.
Arithmetic Progression Formula (Reference Information)
The formula provided, Sn = n/2[2a + (n − 1) × d], calculates the sum of 'n' terms of an arithmetic progression, where:
- Sn = Sum of 'n' terms
- n = Number of terms
- a = First term
- d = Common difference
This formula isn't directly used to find the common difference itself, but it's useful for calculating the sum of the terms in a progression once 'd' is known.
