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 (1x)^{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} = a^{n}(1+x/a)^{n} if < 1
IMPORTANT Binomial theorem formula
(1+ x)^{1} = 1 x +x^{2} –x^{3 }+ . . . + (1)^{r}x^{r}+. . .
(1  x)^{1} = 1+ x +x^{2 }+x^{3} + . . .+ x^{r }+ . . .
(1+ x)^{2} = 1 2x +3x^{2} –4x^{3}+ . . .+ (1)^{r}(r+1)x^{r}+. . .
(1  x)^{2} = 1+ 2x +3x^{2} +4x^{3} + . . .+ (r+1)x^{r}+. . .
(1+x)^{3} = 1 3x +6x^{2} –10x^{3} +. . .+ (1)^{r} x^{r}+. . .
(1x)^{3} = 1+ 3x +6x^{2} +10x^{3} + . . .+ 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 + 3x^{2} + 4x^{3} + …..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 + 3x^{2} + 4x^{3} + … to ∞.