MP Board · Class 10 · Mathematics · Arithmetic ProgressionsExplain the concept of an Arithmetic Progression (AP) in detail. Discuss its general form, how to identify an AP using the common difference, and derive the formula for the nth term of an AP with a detailed explanation of each component.

Introduction to Arithmetic Progression (AP)\nAn Arithmetic Progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a fixed number to the preceding term. This fixed number is called the common difference.

General Form of an AP

• If the first term of an AP is denoted by '$a$' and the common difference is denoted by '$d$', then the AP can be represented in the following general form: $$a, a + d, a + 2d, a + 3d, \dots$$
• Here, the first term is $a_1 = a$, the second term is $a_2 = a + d$, the third term is $a_3 = a + 2d$, and so on.

The Common Difference

• The common difference '$d$' is constant throughout the progression.
• It can be positive, negative, or zero.
• Mathematically, if $a_n$ is the $n$-th term of an AP, the common difference is given by: $$d = a_{n} - a_{n-1}$$ where $n > 1$.

Derivation of the nth Term ($a_n$)\nLet us consider an AP whose first term is $a$ and common difference is $d$:

• First term ($a_1$) = $a = a + (1 - 1)d$
• Second term ($a_2$) = $a + d = a + (2 - 1)d$
• Third term ($a_3$) = $a + 2d = a + (3 - 1)d$
• Fourth term ($a_4$) = $a + 3d = a + (4 - 1)d$ \nObserving this pattern, we can generalize that for any $n$-th term ($a_n$), the multiplier of $d$ is one less than the term number $n$. Therefore, the formula for the $n$-th term of an AP is: $$a_n = a + (n - 1)d$$

Conclusion\nThis formula allows us to find any term of an AP without writing out all the preceding terms, provided we know the first term and the common difference.