Algebra

Algebraic Manipulation

Algebraic Manipulation: Level 3 Challenges

         

Recall that
\[ e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots. \] Then what is the value of \[\large \frac{\frac{2}{3!} + \frac{4}{5!} + \frac{6}{7!} + \cdots}{\frac{2}{1!} + \frac{4}{3!} + \frac{6}{5!} + \cdots}\, ? \]

\(\)
Notation: \(!\) is the factorial notation. For example, \(8! = 1\times2\times3\times\cdots\times8 \).

\[\large \begin{cases} a+b+c=50 \\ 3a+b-c=70 \end{cases} \]

\(a,b\) and \(c\) are positive numbers satisfying the system of equations above.

If the range of \( 5a + 4b + 2c \) is \( (m,n) \), what is \( m+ n \)?

\[\large 2016^{x}+2016^{-x}=3\]

\[\large \sqrt{\frac{2016^{6x}-2016^{-6x}}{2016^{x}-2016^{-x}}} = \, ?\]

\[S = \displaystyle\sqrt{9 - \sqrt{\dfrac{13}{9} + \sqrt{\dfrac{13}{81} - \sqrt{\dfrac{13}{6561} + \sqrt{\dfrac{13}{43046721} - .......}}}}}\]

If \(S = \sqrt{\dfrac{a}{b}}\) where \(a, b\) are both primes, find \(a + b\).

\[\sum_{j=2}^{2016} \sum_{k=1}^{j-1} \dfrac kj = \frac{1}{2}+\frac{1}{3}+\frac{2}{3}+\frac{1}{4}+\frac{2}{4}+\frac{3}{4}+\cdots+\frac{2013}{2016}+\frac{2014}{2016}+\frac{2015}{2016}= \, ? \]

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