“Courage is one step ahead of fear.” – Coleman Young

# Lecture - 6 Chapter 8 Binomial Theorem

NCERT Examples |

**Example 4.** Using binomial theorem, prove that 6^n-5n always leaves remainder 1 when divided by 25.

**Example 6. ** Show that the middle term in the expansion of (1+x)^{2n} is \frac{1.3.5…(2n-1)}{n!} 2^n . x^n, where *n* is a positive integer.

**Example 10.** Find the term independent of *x* in the expansion of \left ( \frac{3}{2}x^2 – \frac{1}{3x} \right )^6.

**Example 11.** If the coefficients of a^{r-1}, a^r \text{ and } a^{r+1} in the expansion of (1+a)^n are in arithmetic progression, prove that n^2-n(4r+1)+4r^2-2=0.

**Example 12.** Show that the coefficient of the middle term in the expansion of (1+x)^{2n} is equal to the sum of the coefficients of two middle terms in the expansion of (1+x)^{2n-1} .

#### Clip - 1

**Example 14.** Find the *r*^{th }term from the end in the expansion of (x+a)^n.

**Example 15.** Find the term independent of *x* in the expansion of \left ( \sqrt[3]{x} +\frac{1}{2 \sqrt[3]{x}} \right )^{18} , x>0.

**Example 16.** The sum of the coefficients of the first three terms in the expansion of \left ( x – \frac{3}{x^2} \right )^m, x \ne 0, *m *being a natural number, is 559. Find the term of the expansion containing x^3.

**Example 17.** If the coefficients of (r-5)^{th} and (2r-1)^{th} terms in the expansion of (1+x)^{34} are equal, find *r*.