Question: Determine the numerical value of the following expression without the use of a calculator:
(100log10(1000100)+n=1∑100(−1)nsin(πn)+1).m=1∏1000cos(πm)21
Solution:
For the logarithm operation:
100log10(1000100)
=100100log101000
=100100log10103
=100(3×100)log1010
since log1010=1;
Then,
∴ 100log10(1000100)=100(3×100)log1010=100(3×100×1)=3
For the cycle operations:
f(n)=n=1∑100(−1)nsin(πn)+1
for all n∈N,sin(πn)=0considering the denominator,for even n,(−1)nsin(πn)+1=1for odd n,(−1)nsin(πn)+1=−1
Therefore, dividing the summation into two parts: i.e. 50 odds and 50 evens, we have:
f(n)=50×((−1)nsin(πn)+1) for denominator n=evens+50×((−1)nsin(πn)+1) for denominator n=odds =(50×1)+(50×−1)=50−50=0
∴ f(n)=n=1∑100(−1)nsin(πn)+1=0 Also, for:
f(m)=m=1∏1000cos(πm)21for all m∈N,cos(πm)21=1
Therefore the product of:
m=1∏1000cos(πm)21=1×1×1×........=1∴ the square root of the product of:m=1∏1000cos(πm)21=1=1
∴ f(m)=m=1∏1000cos(πm)21=1
Therefore, the final answer to the expression is given by:
∴(100log10(1000100)+n=1∑100(−1)nsin(πn)+1).m=1∏1000cos(πm)21
=(3+0)×1=3×1=3
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