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 = T_r+1 = (-1)r x^r

• General term of the series  (1-x)^-n = T_r+1 = x^r 

• 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)^rx^r+. . .

(1 - x)^-1 = 1+ x +x² +x³ + . . .+ x^r + . . .

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

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

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

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

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 ∞.