Find the 10th term of the AP : $2, 7, 12, \ldots$

Step 1: Identify the first term and common difference

In an Arithmetic Progression (AP), the first term is denoted by $a$. From the given sequence, the first term is $2$. The common difference, denoted by $d$, is found by subtracting any term from its succeeding term. Here, the common difference is $7 - 2 = 5$.

Step 2: Identify the number of the term to find

We need to find the 10th term of the AP, so the value of $n$ (the term number) is $10$.

Step 3: Apply the formula for the nth term of an AP

The formula for the $n^{th}$ term of an Arithmetic Progression is $a_n = a + (n-1)d$, where $a_n$ is the $n^{th}$ term, $a$ is the first term, $n$ is the term number, and $d$ is the common difference. This formula is fundamental for finding any term in an AP. (Refer to NCERT Class 10 Mathematics - Arithmetic Progressions, Example 3, page 1).

Step 4: Substitute the values into the formula

Now we substitute the identified values of $a = 2$, $n = 10$, and $d = 5$ into the formula $a_n = a + (n-1)d$ to calculate the 10th term.

Step 5: Calculate the 10th term

Perform the arithmetic operations: first, subtract $1$ from $10$ to get $9$, then multiply $9$ by the common difference $5$ to get $45$. Finally, add this product to the first term $2$ to obtain the 10th term, which is $47$.

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