Final answer:
In this case, the ratio S₃ₙ/Sₙ is equal to 3 * (3n + 1) * a₁.
None of the given options is correct
Step-by-step explanation:
To find the ratio S₃ₙ/Sₙ, we first need to understand the relationship between S₂ₙ and Sₙ.
1. Let's consider an arithmetic progression (AP) with the common difference d and the first term a₁.
2. The sum of the first n terms of this AP, denoted as Sn, can be calculated using the formula: Sn = (n/2)(2a₁ + (n-1)d).
3. According to the given information, S₂ₙ = 2Sₙ. Substituting the respective formulas for S₂ₙ and Sₙ into the equation, we get:
(2n/2)(2a₁ + (2n-1)d) = 2(n/2)(2a₁ + (n-1)d).
4. Simplifying the equation, we have: n(2a₁ + (2n-1)d) = n(2a₁ + (n-1)d).
5. Canceling the common factor n, we get: 2a₁ + (2n-1)d = 2a₁ + (n-1)d.
6. Rearranging the equation, we have: d = a₁.
From step 6, we can conclude that the common difference d of the AP is equal to the first term a₁.
Now, let's find the ratio S₃ₙ/Sₙ using the formula for Sn:
- S₃ₙ = (3n/2)(2a₁ + (3n-1)d).
- Sₙ = (n/2)(2a₁ + (n-1)d).
Dividing S₃ₙ by Sₙ:
S₃ₙ/Sₙ = [(3n/2)(2a₁ + (3n-1)d)] / [(n/2)(2a₁ + (n-1)d)].
Simplifying the expression:
S₃ₙ/Sₙ = (3n/2) * (2a₁ + (3n-1)d) / (n/2) * (2a₁ + (n-1)d).
The factors (n/2) and (2a₁ + (n-1)d) cancel out, leaving:
S₃ₙ/Sₙ = 3n * (2a₁ + (3n-1)d) / n.
Simplifying further:
S₃ₙ/Sₙ = 3 * (2a₁ + (3n-1)d).
Since we know from step 6 that d = a₁, we can substitute this into the equation:
S₃ₙ/Sₙ = 3 * (2a₁ + (3n-1)a₁).
Simplifying:
S₃ₙ/Sₙ = 3 * (2a₁ + (3n-1)a₁) = 3 * (2 + (3n-1))a₁ = 3 * (2 + 3n - 1)a₁.
Finally, simplifying the expression:
S₃ₙ/Sₙ = 3 * (3n + 1)a₁.
From this equation, we can conclude that the ratio S₃ₙ/Sₙ is equal to 3 * (3n + 1) * a₁.
None of the given options is correct