In an arithmetic progression, if $S_{40} = 1030$ and $S_{12} = 57$, then $S_{30} - S_{10}$ is equal to:

Step 1: Define the sum of an arithmetic progression

The sum of the first $n$ terms of an arithmetic progression (AP) is given by the formula $S_n = \frac{n}{2}[2a + (n-1)d]$, where $a$ is the first term and $d$ is the common difference.

Step 2: Formulate equations from given information

We are given $S_{40} = 1030$ and $S_{12} = 57$. Substituting these values into the sum formula, we get two linear equations in terms of $a$ and $d$. Simplifying these equations gives us $2a + 39d = 51.5$ and $2a + 11d = 9.5$.

Step 3: Solve for 'a' and 'd'

To find the values of $a$ and $d$, we subtract equation (2) from equation (1). This eliminates $2a$, allowing us to solve for $d$. Once $d$ is found, we substitute it back into equation (2) to solve for $a$. We find that $d = 1.5$ and $a = -3.5$.

Step 4: Calculate $S_{30}$ and $S_{10}$

Now that we have the values for $a$ and $d$, we can calculate $S_{30}$ and $S_{10}$ using the sum formula. Substituting $n=30$, $a=-3.5$, and $d=1.5$ gives $S_{30} = 547.5$. Similarly, substituting $n=10$, $a=-3.5$, and $d=1.5$ gives $S_{10} = 32.5$.

Step 5: Find the difference $S_{30} - S_{10}$

Finally, we subtract $S_{10}$ from $S_{30}$ to find the required value. The difference is $547.5 - 32.5 = 515$.

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