====== Ch 08: Mathematical Induction and Binomial Theorem ======
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  * Using binomial theorem,expand $\left(\frac{x}{2}-\frac{2}{x^2}\right)$ --- // BISE Gujranwala(2015)//
  * Find the $6$th term in the expansion of $\left( x^2-\frac{3}{2x}\right)$ --- // BISE Gujranwala(2015)//
  * Expand $\left( 8-2x\right)^{-1}$ up to two terms. --- // BISE Gujranwala(2015)//
  * Use binomial theorem to show that $1+\frac{1}{4}+\frac{1.3}{4.8}+\frac{1.3.5}{4.8.12},...=\sqrt{2}$  --- // BISE Gujranwala(2015), BISE Sargodha(2016)//
  * Evaluate $(1.03)^{\frac{1}{3}}$ by binomial theorem upto three places of decimials. --- // BISE Gujranwala(2017)//
  * Find the middle term of $(a+x)$? When $n$ is even. --- // BISE Gujranwala(2017)//
  * Find the term independent of $x$ in the following expansion $(x-\frac{2}{x})^{10}$  --- // BISE Sargodha(2015), BISE Gujranwala(2017)//
  * Show that $n^3-n$ is divisible by $6$ by $n=2,3$  --- // BISE Gujranwala(2017)//
  * Verify the result $4^n>3^n+2^{n-1}$ for $n=2,3$  --- // BISE Sargodha(2015)//
  * Find $13$th term of $x,1,2-x,3-2x,...$ --- // BISE Sargodha(2015)//
  * Expand $(1-2x)^{\frac{1}{3}}$ upto three term. --- // BISE Sargodha(2015)//
  * Sum of the series $-8-3 \frac{1}{2}+1+...{a}_{11} $ --- // BISE Sargodha(2015)//  
  * Find $5$th term in the expansion of $(\frac{3}{2}x-\frac{1}{3x})^{11}$ --- // BISE Sargodha(2015)//
  * Find the term involving $x^{-2}$ in expansion $(x-\frac{2}{x^2})^{13}$ --- // BISE Sargodha(2015)//
  * Expand $(4-3x)^{\frac{1}{2}}$ upto two terms. --- // BISE Sargodha(2016)//
  * Expand $(3a-\frac{x}{3a})^4$ by binomial theorem  --- // BISE Sargodha(2016),BISE Sargodha(2017)//
  * Use mathematical induction to prove the following formula for every positive integer $n$. $1+\frac{1}{2}+\frac{1}{4}+...+\frac{1}{2^{n-1}}=2[1-\frac{1}{2^n}]$ --- // BISE Sargodha(2017), BISE Lahore(2017)//
  * Expand $(8-5x)^{\frac{-2}{3}}$ upto two terms. --- // BISE Sargodha(2017)//
  * If $x$ is so small that bits square and higher powers can be neglected then show that $\frac{1-x}{\sqrt{1+x}} \approx {1-\frac{3}{2}}$ --- // BISE Sargodha(2015)//
  * Prove that $1+5+9+...+(4n-3)=n(2n-1)$ for $n=1$ and $n=2$ --- // BISE Lahore(2017)//
  * Expand upto three terms $(1-x)^{\frac{1}{2}}$ --- // BISE Lahore(2017)//
  * Using binomial theorem, calculate $(0.97)^3$ --- // BISE Lahore(2017)//
  * Expand $(1-2x)^{\frac{1}{3}}$ upto three terms. --- // BISE Lahore(2017)//
  * Use mathematical induction to prove that $1+4+7+...+(3n-2)=\frac{n(3n-1)}{2}$ --- // BISE Lahore(2017)//
  * Find the coefficient of $x^n$ in the expansion of $(1-x+x^2-x^3+...)^2$ --- // FBISE (2016)// 
  * Use principal of mathematical induction to show that $1^2+3^2+5^2+...+(2n-1)^2$ for every positive integer $n$. --- // FBISE (2016)// 
  * Show that the middle term of $(1+x)^{2n}$ is $\frac{1.3.5...(2n-1)}{n!}2^nx^n$ --- // FBISE (2017)// 
  * Expand $\frac{(4+2x)^{\frac{1}{2}}}{2-x}$ upto $4$ terms. --- // FBISE (2017)// 

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{{tag>FSc FSc_Part1 Important_Questions_FSc_1}}