Question #250578

Determine the numerical value of the following expression without the use of a calculator:

 

log10 (1000100)

100

+

X100

n=1

sin(n) + 1

(􀀀1)n

!



vuut

1Y000

m=1

1

cos(m)2


1
Expert's answer
2021-10-19T12:28:48-0400

Question: Determine the numerical value of the following expression without the use of a calculator:



(log10(1000100)100+n=1100sin(πn)+1(1)n).m=110001cos(πm)2(\frac{\log_{10} (1000^{100})} {100} + \sum_{n = 1}^{100}\frac{\sin(\pi n)+1}{(-1)^n}) . \sqrt{\prod_{m=1}^{1000}\frac{1}{\cos (\pi m)^{2}}}



Solution:

For the logarithm operation:

log10(1000100)100\frac{\log_{10} (1000^{100})} {100}

=100log101000100= \frac{100\log_{10} 1000} {100}


=100log10103100= \frac{100\log_{10} 10^3} {100}


=(3×100)log1010100= \frac{(3\times 100)\log_{10} 10} {100}


since log1010=1\log_{10} 10 = 1;

Then,



 log10(1000100)100=(3×100)log1010100=(3×100×1)100=3\therefore\ \frac{\log_{10} (1000^{100})} {100} = \frac{(3\times 100)\log_{10} 10} {100} = \frac{(3\times 100 \times 1)} {100} = 3


For the cycle operations:

f(n)=n=1100sin(πn)+1(1)nf(n) = \sum_{n = 1}^{100}\frac{\sin(\pi n)+1}{(-1)^n}



for all nN,sin(πn)=0considering the denominator,for even n,sin(πn)+1(1)n=1for odd n,sin(πn)+1(1)n=1for \ all \ {n \in \mathbb{N}}, \sin(\pi n) = 0 \\ considering \ the \ denominator, \\ for \ even \ n, \frac{\sin(\pi n) + 1} {(-1)^n} =1 \\ for \ odd \ n, \frac{\sin(\pi n) + 1} {(-1)^n} = -1

Therefore, dividing the summation into two parts: i.e. 50 odds and 50 evens, we have:

f(n)=50×(sin(πn)+1(1)n) for denominator n=evens+50×(sin(πn)+1(1)n) for denominator n=odds =(50×1)+(50×1)=5050=0f(n) = \\ 50 \times (\frac{\sin(\pi n) + 1} {(-1)^n}) \ for\ denominator\ n = evens \\ + 50 \times (\frac{\sin(\pi n) + 1} {(-1)^n}) \ for\ denominator\ n = odds \\\ = (50 \times 1) + (50\times -1) = 50 - 50 = 0

 f(n)=n=1100sin(πn)+1(1)n=0\therefore \ f(n) = \sum_{n = 1}^{100}\frac{\sin(\pi n)+1}{(-1)^n} = 0

Also, for:

f(m)=m=110001cos(πm)2f(m) = \sqrt{\prod_{m=1}^{1000}\frac{1}{\cos (\pi m)^{2}}}

for all mN,1cos(πm)2=1for \ all \ {m \in \mathbb{N}}, \frac{1} {\cos(\pi m)^2} = 1

Therefore the product of:


m=110001cos(πm)2=1×1×1×........=1 the square root of the product of:m=110001cos(πm)2=1=1\prod_{m=1}^{1000}\frac{1}{\cos (\pi m)^{2}} = 1 \times 1 \times 1 \times........ = 1 \\ \\ \therefore \ the \ square \ root \ of \ the \ product \ of: \\ \sqrt{\prod_{m=1}^{1000}\frac{1}{\cos (\pi m)^{2}}} = \sqrt{1} = 1


 f(m)=m=110001cos(πm)2=1\therefore \ f(m) = \sqrt{\prod_{m=1}^{1000}\frac{1}{\cos (\pi m)^{2}}} = 1



Therefore, the final answer to the expression is given by:



(log10(1000100)100+n=1100sin(πn)+1(1)n).m=110001cos(πm)2\therefore (\frac{\log_{10} (1000^{100})} {100} + \sum_{n = 1}^{100}\frac{\sin(\pi n)+1}{(-1)^n}) . \sqrt{\prod_{m=1}^{1000}\frac{1}{\cos (\pi m)^{2}}}




=(3+0)×1= (3 + 0) \times 1=3×1= 3 \times 1=3= 3




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