Question

The first term of an AP is $p$ and the common difference is $q$, then its 10th term is :

• A. $q - 9p$
• B. $p - 9q$
• C. $p + 9q$
• D. $2p + 9q$

Hint:

For any AP, the coefficient of $d$ in the $n$-th term is always $(n-1)$.
Thus, the 5th term is $a + 4d$, the 10th term is $a + 9d$, and the 100th term is $a + 99d$. This simple rule avoids calculation errors.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The first term ($a$) of an Arithmetic Progression (AP) is given as $p$, and its common difference ($d$) is given as $q$. We need to find the value of its 10th term ($a_{10}$).

Step 2: Key Formula or Approach:
The general formula for the $n$-th term ($a_n$) of an Arithmetic Progression is:
\[ a_n = a + (n - 1)d \]
where:
- $a$ is the first term,
- $d$ is the common difference,
- $n$ is the position of the term in the sequence.

Step 3: Detailed Explanation:

• 1. Identify the given parameters from the question:
- First term, $a = p$
- Common difference, $d = q$
- The term index we want to find, $n = 10$

• 2. Write down the formula for the $n$-th term:
\[ a_n = a + (n - 1)d \]

• 3. Substitute the values of $a$, $d$, and $n$ into the formula:
\[ a_{10} = p + (10 - 1)q \]

• 4. Simplify the expression inside the parentheses:
\[ a_{10} = p + 9q \]


Step 4: Final Answer:
The 10th term of the AP is $p + 9q$, which corresponds to option (C).