“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 \( (1+x)^{2n-1} \).

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**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*.