Concept 5.2
The Common Difference and Its Sign
Source: NCERT Mathematics, Class 108 minute lesson
Formal statement
For an AP a₁, a₂, a₃, …, the common difference is d = a₂ − a₁ = a₃ − a₂ = …. If d > 0 the terms increase, if d < 0 they decrease, and if d = 0 every term is the same.
The intuition
d is a direction, not just a size. In 10, 7, 4, 1, each step moves 3 to the left, so d = −3.
The sign comes from the order you subtract in. Always take a term minus the one before it. Reverse the order and you get the right size pointing the wrong way.
The attention
Look at the order of the subtraction: the later term comes first.
d = a₂ − a₁
Look hereNot here
- Look here
- The later term. Start from the later term. That is what makes d point the way the list actually moves.
- Not here
- The earlier term. Starting from the first term because it comes first gives a₁ − a₂: the same size with the opposite sign.
- What it teaches
- A difference has a direction. The order you subtract in is part of the answer.