The nth term of an AP with first term a and common difference d is:

An Arithmetic Progression (AP) is a sequence of numbers where the difference between consecutive terms is constant. This constant is called the common difference, denoted by $d$. Let the terms of the AP be $T_1, T_2, T_3, \dots, T_n$. 1. **First term ($T_1$):** By definition, $T_1 = a$. 2. **Second term ($T_2$):** We add the common difference once to the first term: $T_2 = a + d$. 3. **Third term ($T_3$):** We add $d$ to the second term: $T_3 = (a + d) + d = a + 2d$. 4. **Fourth term ($T_4$):** Similarly, $T_4 = a + 3d$. **Observation:** Notice the relationship between the term number ($n$) and the coefficient of $d$: - For $n=1$, coefficient is $0$ ($1-1$) - For $n=2$, coefficient is $1$ ($2-1$) - For $n=3$, coefficient is $2$ ($3-1$) Following this pattern, for the **$n$-th term ($T_n$)**, the common difference $d$ is added $(n-1)$ times to the first term $a$. Thus, the formula is: $$T_n = a + (n - 1)d$$