Question

$\displaystyle \rm \bigg( \dfrac{\log_{10}(1000^{100} )}{100} + \sum_{n = 1}^{100} \dfrac{ \sin(\pi n) + 1}{ { (- 1)}^{n} } \bigg) \sqrt{ \prod_{m = 1}^{1000} \dfrac{ 1}{ \cos(\pi m)^{2} } }$​

Answer 1

Appropriate Question :- Evaluate

$\displaystyle \rm \bigg( \dfrac{\log_{10}(1000^{100} )}{100} + \sum_{n = 1}^{100} \dfrac{ \sin(\pi n) + 1}{ { (- 1)}^{n} } \bigg) \sqrt{ \prod_{m = 1}^{1000} \dfrac{ 1}{( \cos(\pi m))^{2} } }$

$\large\underline{\sf{Solution-}}$

Given expression is

$\displaystyle \rm \bigg( \dfrac{\log_{10}(1000^{100} )}{100} + \sum_{n = 1}^{100} \dfrac{ \sin(\pi n) + 1}{ { (- 1)}^{n} } \bigg) \sqrt{ \prod_{m = 1}^{1000} \dfrac{ 1}{ (\cos(\pi m))^{2} } }$

Let we consider,

$\rm \: \displaystyle \rm \dfrac{\log_{10}(1000^{100} )}{100}$

We know,

$\boxed{ \rm{ \: log_{a}( {x}^{y} ) = y \: log_{a}(x) \: }}$

So, using this, we get

$\rm \: = \: \displaystyle \rm \dfrac{100 \: \log_{10}1000}{100}$

$\rm \: = \: log_{10}1000$

$\rm \: = \: log_{10} {10}^{3}$

We know,

$\boxed{ \rm{ \: log_{a}( {a}^{x} ) = x \: }}$

So, using this, we get

$\rm \: = \: 3$

Thus,

$\rm\implies \:\boxed{ \rm{ \:\displaystyle \rm \dfrac{\log_{10}(1000^{100} )}{100} = 3 \: \: }}$

Now, Consider

$\displaystyle \rm \sum_{n = 1}^{100} \dfrac{ \sin(\pi n) + 1}{ { (- 1)}^{n} }$

can be rewritten as

$\rm \: = \displaystyle \rm \sum_{n = 1}^{100} \dfrac{ \sin(n\pi ) + 1}{ { (- 1)}^{n} }$

We know,

$\boxed{ \rm{ \:sin \: (n\pi) = 0 \: \forall \: n \: \in \: N \: }}$

So, above expression can be rewritten as

$\rm \: = \displaystyle \rm \sum_{n = 1}^{100} \dfrac{ 0+ 1}{ { (- 1)}^{n} }$

$\rm \: = \displaystyle \rm \sum_{n = 1}^{100} \dfrac{1}{ { (- 1)}^{n} }$

$\rm \: = \: \underbrace{- 1 + 1 - 1 + 1 - 1 + \cdots + 1} \\ \: \: \: \: \: \: \: 100 \: terms$

$\rm \: = \: 0$

Hence,

$\rm\implies \:\boxed{ \rm{ \:\displaystyle \rm \sum_{n = 1}^{100} \dfrac{ \sin(\pi n) + 1}{ { (- 1)}^{n} } = 0 \: \: }}$

Now, Consider

$\displaystyle \rm \sqrt{ \prod_{m = 1}^{1000} \dfrac{ 1}{ (\cos(\pi m))^{2} } }$

can be rewritten as

$\displaystyle \rm = \: \sqrt{ \prod_{m = 1}^{1000} \dfrac{ 1}{ (\cos(m\pi ))^{2} } }$

We know,

$\boxed{ \rm{ \:cos\: (n\pi) = {( - 1)}^{n} \: \: \: \: \: \forall \: n \: \in \: N \: }}$

So, using this, we get

$\displaystyle \rm = \: \sqrt{ \prod_{m = 1}^{1000} \dfrac{ 1}{ ( - 1)^{2m} } }$

$\displaystyle \rm = \: \sqrt{ \prod_{m = 1}^{1000} \dfrac{ 1}{ 1 } }$

$\displaystyle \rm = \: \sqrt{ \prod_{m = 1}^{1000} 1 }$

$\rm \: = \: \sqrt{ \underbrace{1 \times 1 \times 1 \times \cdots \times 1}} \\ \: \: \: \: \: \: \: \: 1000 \: terms$

$\rm \: = \: \sqrt{1}$

$\rm \: = \: 1$

Thus,

$\rm\implies \:\boxed{ \rm{ \:\displaystyle \rm \sqrt{ \prod_{m = 1}^{1000} \dfrac{ 1}{ (\cos(\pi m))^{2} } } = 1 \: }}$

Now, Consider the given expression

$\displaystyle \rm \bigg( \dfrac{\log_{10}(1000^{100} )}{100} + \sum_{n = 1}^{100} \dfrac{ \sin(\pi n) + 1}{ { (- 1)}^{n} } \bigg) \sqrt{ \prod_{m = 1}^{1000} \dfrac{ 1}{ (\cos(\pi m))^{2} } }$

On substituting the values evaluated above, we get

$\rm \: = \: (3 + 0) \times 1$

$\rm \: = \: 3$

Hence,

$\boxed{ \rm{ \:\displaystyle \rm \bigg( \dfrac{\log_{10}(1000^{100} )}{100} + \sum_{n = 1}^{100} \dfrac{ \sin(\pi n) + 1}{ { (- 1)}^{n} } \bigg) \sqrt{ \prod_{m = 1}^{1000} \dfrac{ 1}{( \cos(\pi m))^{2} } } = 3}}$

$\rule{190pt}{2pt}$

Additional Information :-

$\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{ \red{More \: Formulae}}}} \\ \\ \bigstar \: \bf{ log_{x}(x) = 1}\\ \\ \bigstar \: \bf{ log_{x}( {x}^{y} ) = y}\\ \\ \bigstar \: \bf{ log_{ {x}^{z} }( {x}^{w} ) = \dfrac{w}{z} }\\ \\ \bigstar \: \bf{ log_{a}(b) = \dfrac{logb}{loga} }\\ \\ \bigstar \: \bf{ {e}^{logx} = x}\\ \\ \bigstar \: \bf{ {e}^{ylogx} = {x}^{y}}\\ \\ \bigstar \: \bf{log1 = 0}\\ \\ \end{array} }}\end{gathered}\end{gathered}\end{gathered}$