Understanding the Given Problem
The sum of the first n terms of two arithmetic progressions (APs) is given in the ratio (7n - 5) : (5n + 17). We need to determine which terms of these APs are equal to each other.
Step 1: Sum of n Terms of APs
- The sum of the first n terms of an AP is given by the formula:
Sn = n/2 * [2a + (n - 1)d]
- Let the first AP have the first term a1 and common difference d1.
- Let the second AP have the first term a2 and common difference d2.
Step 2: Setting Up the Equation
- For the first AP: S1n = n/2 * [2a1 + (n - 1)d1]
- For the second AP: S2n = n/2 * [2a2 + (n - 1)d2]
- The ratio of sums can be written as:
(n/2 * [2a1 + (n - 1)d1]) / (n/2 * [2a2 + (n - 1)d2]) = (7n - 5) / (5n + 17)
Step 3: Equating the Terms
After simplifying, we can equate the coefficients of n and the constant terms from both sides of the equation.
Step 4: Finding Equal Terms
To find the equal terms of the APs:
- The nth term of the first AP: T1n = a1 + (n - 1)d1
- The nth term of the second AP: T2n = a2 + (n - 1)d2
- Set T1n = T2n for specific n values.
Conclusion: Identifying Equal Terms
The specific value of n can be determined by solving the equations obtained from the ratio. This will yield the terms of the two APs that are equal, allowing us to find the relationship between the sequences based on their initial conditions and differences.