Key Concepts and Formulas

• The sum of the first $n$ terms of an arithmetic progression (AP) with first term $a$ and common difference $d$ is given by: $S_n = \frac{n}{2}[2a + (n-1)d]$
• A system of linear equations can be solved using methods like substitution or elimination to find the values of unknown variables.

Step-by-Step Solution

Step 1: Formulate equations from the given information. We are given the sum of the first 20 terms ($S_{20}$) and the sum of the first 10 terms ($S_{10}$) of an arithmetic progression. We will use the formula for $S_n$ to create two equations with the first term ($a$) and the common difference ($d$) as unknowns.

For $S_{20} = 790$: $S_{20} = \frac{20}{2}[2a + (20-1)d] = 10[2a + 19d]$ Given $S_{20} = 790$, we have: $10(2a + 19d) = 790$ Divide by 10: $2a + 19d = 79 \quad \text{.... (1)}$

For $S_{10} = 145$: $S_{10} = \frac{10}{2}[2a + (10-1)d] = 5[2a + 9d]$ Given $S_{10} = 145$, we have: $5(2a + 9d) = 145$ Divide by 5: $2a + 9d = 29 \quad \text{.... (2)}$

Step 2: Solve the system of linear equations to find the values of 'a' and 'd'. We have a system of two linear equations:

1. $2a + 19d = 79$
2. $2a + 9d = 29$

We can solve this system by subtracting equation (2) from equation (1) to eliminate $a$. $(2a + 19d) - (2a + 9d) = 79 - 29$ $10d = 50$ $d = \frac{50}{10}$ $d = 5$

Now, substitute the value of $d=5$ into equation (2) to find $a$. $2a + 9(5) = 29$ $2a + 45 = 29$ $2a = 29 - 45$ $2a = -16$ $a = \frac{-16}{2}$ $a = -8$

So, the first term of the AP is $a = -8$ and the common difference is $d = 5$.

Step 3: Calculate $S_{15}$ and $S_5$ using the found values of 'a' and 'd'. Now we need to find $S_{15}$ and $S_5$ using the formula $S_n = \frac{n}{2}[2a + (n-1)d]$.

For $S_{15}$: $S_{15} = \frac{15}{2}[2(-8) + (15-1)(5)]$ $S_{15} = \frac{15}{2}[-16 + (14)(5)]$ $S_{15} = \frac{15}{2}[-16 + 70]$ $S_{15} = \frac{15}{2}[54]$ $S_{15} = 15 \times 27$ $S_{15} = 405$

For $S_5$: $S_5 = \frac{5}{2}[2(-8) + (5-1)(5)]$ $S_5 = \frac{5}{2}[-16 + (4)(5)]$ $S_5 = \frac{5}{2}[-16 + 20]$ $S_5 = \frac{5}{2}[4]$ $S_5 = 5 \times 2$ $S_5 = 10$

Step 4: Calculate the difference $S_{15} - S_5$. $S_{15} - S_5 = 405 - 10$ $S_{15} - S_5 = 395$

Common Mistakes & Tips

• Algebraic Errors: Be extremely careful when solving the system of equations for $a$ and $d$, especially with signs.
• Formula Application: Ensure the correct formula for $S_n$ is used and that $n$ is substituted accurately in each term of the formula.
• Calculation Accuracy: Double-check all arithmetic calculations, especially multiplications and additions/subtractions, to avoid compounding errors.

Summary

The problem requires us to find the difference between the sum of the first 15 terms and the sum of the first 5 terms of an arithmetic progression, given the sums of the first 20 and 10 terms. We successfully formulated two linear equations using the given information and the formula for $S_n$. By solving this system, we determined the first term ($a = -8$) and the common difference ($d = 5$). Subsequently, we calculated $S_{15}$ and $S_5$ using these values and found their difference. The calculated difference is 395.

The final answer is $\boxed{395}$.