An arithmetic progression (AP) is a sequence in which each term increases by a constant difference \(d \). That is, the difference between any two consecutive terms remains the same. Based on this property, we can easily find the \(n^{th}\) term of an arithmetic progression.
Let \(a_1\) be the first term and \(d\) be the common difference. Then the \(n^{th}\) term can be found as follows:
\( \begin{align*} a_1 &= \text{First term}, \\ a_2 &= a_1 + d, \\ a_3 &= a_2 + d = (a_1 + d) + d = a_1 + 2d, \\ a_4 &= a_3 + d = (a_1 + 2d) + d = a_1 + 3d, \\ &\;\;\vdots \\ a_n &= a_{n-1} + d = a_1 + (n-1)d \\ \end{align*} \)Thus, the \(n^{th}\) term \((a_n)\) of an AP is: \( \boxed{a_n = a_1 + (n – 1)d} \)
Here is an example:
If \(a_1 = 15 \) and \( d = 5 \), using the above formula, we can find \(a_{10} \).
\(a_{10} = a_1 + (10-1)d = 15 + 9 \times 5 = 15+45 = 60 \)