# First Year Math Mathematical Induction and Binomial Theorem

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$Example 4: Evaluate \sqrt[3]{30} correct to three places of decimal.$

$5. Expand the following in descending powers of x :i) \left(x^{2}+x-1\right)^{3}$

$Use mathematical induction to prove the following formulae for every positive integer n . 25. x+1 is a factor of x^{2 n}-1 ;(x \neq-1)$

$3. Find the coefficient of x^{n} in the expansion ofv) \left(1-x+x^{2}-x^{3}+\ldots\right)^{2}$

$Example 5: Show that: \left(\begin{array}{l}n \\ 1\end{array}\right)+2\left(\begin{array}{l}n \\ 2\end{array}\right)+3\left(\begin{array}{l}n \\ 3\end{array}\right)+\ldots .+n\left(\begin{array}{l}n \\ n\end{array}\right)=n .2^{n-1}$