To find the 10th term of the arithmetic progression, let's break down the problem step by step.

Step 1: Formula for the sum of the first $n$ terms

The sum of the first $n$ terms of an arithmetic progression (AP) is given by the formula:

$S_n = \frac{n}{2} \left( 2a + (n-1)d \right)$ where:

• $S_n$ is the sum of the first $n$ terms,
• $a$ is the first term,
• $d$ is the common difference,
• $n$ is the number of terms.

However, the sum of the first $n$ terms is already given as:

$S_n = n^2 - 3n$

Step 2: Deriving the $n$-th term

The $n$-th term of an AP is given by:

$T_n = S_n - S_{n-1}$

We will find $T_n$ by subtracting the sum of the first $n-1$ terms from the sum of the first $n$ terms.

Step 2.1: Find $S_n$ and $S_{n-1}$

We already have the sum of the first $n$ terms:

$S_n = n^2 - 3n$

Now, for $S_{n-1}$, replace $n$ with $n-1$ in the formula:

$S_{n-1} = (n-1)^2 - 3(n-1)$

Simplify the expression for $S_{n-1}$:

$S_{n-1} = (n^2 - 2n + 1) - 3n + 3$ $S_{n-1} = n^2 - 5n + 4$

Step 2.2: Find the $n$-th term

Now, we can find the $n$-th term by subtracting $S_{n-1}$ from $S_n$:

$T_n = S_n - S_{n-1}$ $T_n = (n^2 - 3n) - (n^2 - 5n + 4)$ $T_n = n^2 - 3n - n^2 + 5n - 4$ $T_n = 2n - 4$

Step 3: Find the 10th term

Substitute $n = 10$ into the expression for $T_n$:

$T_{10} = 2(10) - 4$ $T_{10} = 20 - 4 = 16$

So, the 10th term of the arithmetic progression is 16.

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Would you like more details or have any questions about the solution? Here are 5 related questions to explore further:

1. How would you derive the first term of this arithmetic progression?
2. Can you calculate the common difference $d$ of the arithmetic progression?
3. What is the general form of the $n$-th term for an arithmetic progression?
4. How do you calculate the sum of the first 100 terms of an arithmetic progression?
5. How would the formula for the sum of an arithmetic progression change if the sum was $n^2 - 2n$ instead of $n^2 - 3n$?

Tip: To find the common difference $d$ of an arithmetic progression, you can calculate the difference between consecutive terms or use the formula $d = \frac{T_n - T_{n-1}}{n - (n-1)}$.