# Binomial theorem formula

## BINOMIAL THEOREM FOR ANY INDEX

(1+x)n = 1+ nx + + . . . +

### Observations of Binomial Theorem

• Expansion is valid only when –1 <x <1

• General term of the series  (1+x)-n = Tr+1 = (-1)r xr

• General term of the series  (1-x)-n = Tr+1 = x

• If first term is not 1, then make first term unity in the following way: Check out Maths Formulas and NCERT Solutions for class 12 Maths prepared by Physics Wallah.

(a+ x)n = an(1+x/a)n  if < 1

## IMPORTANT Binomial theorem formula

(1+ x)-1 = 1- x +x2 –x3 + . . . + (-1)rxr+. . .

(1 - x)-1 = 1+ x +x2 +x3 + . . .+ xr + . . .

(1+ x)-2 = 1- 2x +3x2 –4x3+ . . .+ (-1)r(r+1)xr+. . .

(1 - x)-2 = 1+ 2x +3x2 +4x3 + . . .+ (r+1)xr+. . .

(1+x)-3 = 1- 3x +6x2 –10x3 +. . .+ (-1)r xr+. . .

(1-x)-3 = 1+ 3x +6x2 +10x3 + . . .+ xr+. . .

In general coefficient of xr in (1 – x)– n is n + r –1Cr .

(1 – x)–p/q = 1 + + …….

(1 + x)–p/q = 1 – – …….

(1 + x)p/q = 1 + + …….

(1 – x)p/q = 1 – – …….

Example :    If –1 < x < 1, show that (1 –x)-2 = 1 + 2x + 3x2 + 4x3 + …..to ∞.

Detail Explanation :  We know that if n is a negative integer or fraction

(1+x)n= 1 +

Provided –1 < x < 1

Putting n = -2 and –x in place of  x, we get

(1+x)2 = 1 +

= 1 + 2x + 3x2 + 4x3 + … to ∞.