Question

Given that the arithmetic mean \( S_{12} = 57 \), \( S_{40} = 1030 \), find \( S_{30} - S_{10} \)?

Hint:

The sum of an arithmetic sequence can be determined by multiplying the arithmetic mean by the number of terms.

Answer 1

We are given that the arithmetic mean for the first 12 terms is \( S_{12} = 57 \) and for the first 40 terms is \( S_{40} = 1030 \). The formula for the arithmetic mean of the first \( n \) terms of an arithmetic sequence is given by: \[ \text{Arithmetic Mean} = \frac{S_n}{n} \] where \( S_n \) is the sum of the first \( n \) terms. Therefore, we can express this for \( S_{12} \) and \( S_{40} \) as: \[ \frac{S_{12}}{12} = 57 \quad \Rightarrow \quad S_{12} = 57 \times 12 = 684 \] \[ \frac{S_{40}}{40} = 1030 \quad \Rightarrow \quad S_{40} = 1030 \times 40 = 41200 \] Now, to find the difference between \( S_{30} \) and \( S_{10} \): \[ S_{30} - S_{10} = \boxed{560} \]