Arithmetic Progressions: Definition & General Form

Definition: Constant Common Difference (d)

In a mathematical sequence, a constant common difference (often denoted as ‘d’) is the value added or subtracted to each term to obtain the next term. This is a fundamental concept in arithmetic sequences.

An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is what we call the ‘common difference,’ represented by the variable ‘d’. If ‘d’ is positive, the sequence increases; if ‘d’ is negative, the sequence decreases; and if ‘d’ is zero, all the terms in the sequence are the same.

Formulae

General Form: $a, a+d, a+2d, a+3d, …$ where ‘a’ is the first term.
Formula for the nth term: $a_n = a + (n-1)d$ where $a_n$ is the nth term.
Formula for the sum of the first n terms (Sn): $S_n = \frac{n}{2}[2a + (n-1)d]$ or $S_n = \frac{n}{2}(a + a_n)$

Examples

Example-1: Consider the sequence 2, 5, 8, 11, … The common difference, d, is 3 because 5-2 = 8-5 = 11-8 = 3. The general form is defined as $a = 2$, $d=3$, therefore the sequence follows the format $2, 2+3, 2+2\times3, 2+3\times3,…$

Example-2: Consider the sequence 10, 7, 4, 1, … The common difference, d, is -3 because 7-10 = 4-7 = 1-4 = -3. The general form is defined as $a = 10$, $d = -3$, therefore the sequence follows the format $10, 10+(-3), 10+2(-3), 10+3(-3),…$

Common mistakes by students

Incorrectly Identifying ‘a’: Students often struggle to identify the first term correctly, sometimes confusing it with another term in the sequence.
Calculating ‘d’ Wrongly: Not subtracting consecutive terms in the correct order (e.g., $a_n – a_{n+1}$ instead of $a_{n+1} – a_n$) leads to the wrong ‘d’ value.
Misapplying the Formula: Using the wrong formula or misinterpreting the values of ‘n’, ‘a’, and ‘d’ in the formulas for the nth term or the sum of the series.

Real Life Application

Arithmetic sequences and the concept of a constant common difference are prevalent in everyday scenarios.

Linear Growth or Decay: Modeling situations with a constant rate of change, such as the increasing amount of interest earned on a simple interest investment, or the decreasing value of a car over time (depreciation).
Creating a Schedule or Pattern: Planning a workout routine where you incrementally increase the number of reps or minutes you exercise each day. This forms an arithmetic sequence.
Financial Planning: Calculating the total cost of payments in a loan with fixed installments.

Fun Fact

The concept of arithmetic sequences dates back to ancient times! Early mathematicians used these sequences to solve practical problems in areas like agriculture and trade.