# Lecture 6 Binomial Theorem Examples Class 11 Maths

“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 rth term from the end in the expansion of $$(x+a)^n$$.

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

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.