Lesson 2 of 10 · 6 min
Patterns in small powers
NCERT §7.2
The stall's helper writes (a + b)⁰ up to (a + b)⁴ on a whiteboard, one under another. Before any rule is given, three patterns can be spotted just by looking.
The lesson in notes
In short
Written out, (a + b)⁰ = 1 (for a + b ≠ 0), (a + b)¹ = a + b, (a + b)² = a² + 2ab + b², (a + b)³ = a³ + 3a²b + 3ab² + b³, and (a + b)⁴ = (a + b)³(a + b) = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴.
Pattern 1: the number of terms is one more than the index. (a + b)⁴ has 5 terms.
Pattern 2: from each term to the next, the power of a falls by 1 and the power of b rises by 1.
Pattern 3: in every term the two powers add up to the index. In 6a²b² the powers are 2 and 2, and 2 + 2 = 4.
The numbers in front, 1, 4, 6, 4, 1 for index 4, are the coefficients. They are the only part that still needs a rule; the powers are fixed by patterns 2 and 3.