# Ch 08: Mathematical Induction and Binomial Theorem

• 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 $1+\frac{1}{4}+\frac{1.3}{4.8}+\frac{1.3.5}{4.8.12},\ldots=\sqrt{2}$ — 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 $x,1,2-x,3-2x,\ldots$ — BISE Sargodha(2015)
• Expand $(1-2x)^{\frac{1}{3}}$ upto three term. — BISE Sargodha(2015)
• Sum of the series $-8-3 \frac{1}{2}+1+\ldots{a}_{11}$ — 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 $n$. $1+\frac{1}{2}+\frac{1}{4}+\ldots+\frac{1}{2^{n-1}}=2[1-\frac{1}{2^n}]$ — 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 $1+5+9+\ldots+(4n-3)=n(2n-1)$ 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 $1+4+7+\ldots+(3n-2)=\frac{n(3n-1)}{2}$ — BISE Lahore(2017)
• Find the coefficient of $x^n$ in the expansion of $(1-x+x^2-x^3+\ldots)^2$ — FBISE (2016)
• Use principal of mathematical induction to show that $1^2+3^2+5^2+\ldots+(2n-1)^2$ for every positive integer $n$. — FBISE (2016)
• Show that the middle term of $(1+x)^{2n}$ is $\frac{1.3.5\ldots(2n-1)}{n!}2^nx^n$ — FBISE (2017)
• Expand $\frac{(4+2x)^{\frac{1}{2}}}{2-x}$ upto $4$ terms. — FBISE (2017)
• fsc-part1-ptb/important-questions/ch08-mathematical-induction-and-binomial-theorem