Arithmetic Progression Class 10 Maths One-Shot Chapter 5 CBSE Class 10 Mid Term Exams

Arithmetic Progressions: The Mathematics of Constant Difference

An Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is always constant. This constant difference, denoted by d, is called the common difference. For example, the sequence 2, 5, 8, 11, 14, ... is an AP with first term a = 2 and common difference d = 3. Arithmetic progressions appear naturally in many real-world situations — the rungs of a ladder are equally spaced, the salary increments of an employee may follow a fixed annual increase, and the number of seats in successive rows of an auditorium often increase by a fixed amount. Understanding APs is essential for the CBSE Class 10 board examination and for higher mathematics including series, calculus, and financial mathematics.

The general form of an AP is a, a + d, a + 2d, a + 3d, ..., where a is the first term and d is the common difference. The nth term (also called the general term) is given by the formula: an = a + (n − 1)d. This formula allows you to find any term in the sequence without listing all the preceding terms. For example, the 20th term of the AP 3, 7, 11, 15, ... is a20 = 3 + (20 − 1)×4 = 3 + 76 = 79. The formula also works in reverse — if you know a specific term and want to find its position, you can solve for n. The common difference can be positive (increasing AP), negative (decreasing AP), or zero (constant sequence). To check whether a given sequence is an AP, compute the difference between successive terms; if all differences are equal, the sequence is an AP.

The sum of the first n terms of an AP is given by two equivalent formulas: Sn = n/2 [2a + (n − 1)d] or Sn = n/2 (a + l), where l = an is the last term. The first formula is used when you know a, d, and n; the second is convenient when you know the first and last terms. These formulas are derived by writing the sum forward and backward and adding them term by term, which doubles the sum but produces n identical terms equal to (a + l). The sum formula is widely used in solving word problems — finding the total distance covered by a freely falling body in n seconds (where distances in successive seconds form an AP with d = g), calculating total savings over months with fixed increment, or finding the sum of odd numbers, even numbers, or multiples of a given number. An important property is that if a given number of terms are selected from an AP at regular intervals, those selected terms also form an AP. Board problems frequently ask students to find the nth term given conditions, determine which term equals a given value, find the sum of a specific number of terms, or solve word problems involving APs in practical contexts.

• An AP has the form a, a+d, a+2d, ... with constant common difference d between consecutive terms.
• The nth term formula: an = a + (n − 1)d — find any term without listing the whole sequence.
• Sum of first n terms: Sn = n/2 [2a + (n−1)d] = n/2 (a + l), where l is the last term.
• If three terms a, b, c are in AP, then 2b = a + c; for selecting terms in AP, use a−d, a, a+d for three terms.
• Applications include fixed salary increments, total distance in free fall, and sums of natural number series.