Some Deductions from Binomial Theorem

1. total number of terms in the expansion = n+1

2. The sum of indices of x and a in each term is n.

3. the coefficients of terms equidistant from the beginning and the end are equal.

4. (x-a)ⁿ
= (r = 0 to n)Σ ((-1) ^r_*ⁿC_rx^n-ra^r

The terms in the expansion of(x-a)ⁿ are alternatively positive and negative, the last term is positive or negative according as n is even or odd.

5. (1+x) ⁿ = (r = 0 to n)Σ ⁿC_rx^r

you get it by putting x =1 and a = x in the expression for (x+a)ⁿ.

6. (x+1) ⁿ = (r = 0 to n)Σ ⁿC_rx^n-r

7. (1-x) ⁿ = (r = 0 to n)Σ(-1)^r_* ⁿC_rx^r

8. (x+a) ⁿ +(x-a) ⁿ = 2[ⁿC₀xⁿa⁰ +
ⁿC₂x^n-2a² +
ⁿC₄x^n-4a⁴+ ...]

9. General term in a binomial expansion

(r+1) term in (x+a) ⁿ
= ⁿC_rx^n-ra^r