Question: Determine the numerical value of the following expression without the use of a calculator:
( log 10 ( 100 0 100 ) 100 + ∑ n = 1 100 sin ( π n ) + 1 ( − 1 ) n ) . ∏ m = 1 1000 1 cos ( π m ) 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}}} ( 100 log 10 ( 100 0 100 ) + n = 1 ∑ 100 ( − 1 ) n sin ( πn ) + 1 ) . m = 1 ∏ 1000 cos ( πm ) 2 1
Solution:
For the logarithm operation:
log 10 ( 100 0 100 ) 100 \frac{\log_{10} (1000^{100})} {100} 100 log 10 ( 100 0 100 )
= 100 log 10 1000 100 = \frac{100\log_{10} 1000} {100} = 100 100 l o g 10 1000
= 100 log 10 1 0 3 100 = \frac{100\log_{10} 10^3} {100} = 100 100 l o g 10 1 0 3
= ( 3 × 100 ) log 10 10 100 = \frac{(3\times 100)\log_{10} 10} {100} = 100 ( 3 × 100 ) l o g 10 10
since log 10 10 = 1 \log_{10} 10 = 1 log 10 10 = 1 ;
Then,
∴ log 10 ( 100 0 100 ) 100 = ( 3 × 100 ) log 10 10 100 = ( 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 ∴ 100 log 10 ( 100 0 100 ) = 100 ( 3 × 100 ) log 10 10 = 100 ( 3 × 100 × 1 ) = 3
For the cycle operations:
f ( n ) = ∑ n = 1 100 sin ( π n ) + 1 ( − 1 ) n f(n) = \sum_{n = 1}^{100}\frac{\sin(\pi n)+1}{(-1)^n} f ( n ) = n = 1 ∑ 100 ( − 1 ) n sin ( πn ) + 1
f o r a l l n ∈ N , sin ( π n ) = 0 c o n s i d e r i n g t h e d e n o m i n a t o r , f o r e v e n n , sin ( π n ) + 1 ( − 1 ) n = 1 f o r o d d n , sin ( π n ) + 1 ( − 1 ) n = − 1 for \ 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 f or a ll n ∈ N , sin ( πn ) = 0 co n s i d er in g t h e d e n o mina t or , f or e v e n n , ( − 1 ) n s i n ( πn ) + 1 = 1 f or o dd n , ( − 1 ) n s i n ( πn ) + 1 = − 1
Therefore, dividing the summation into two parts: i.e. 50 odds and 50 evens, we have:
f ( n ) = 50 × ( sin ( π n ) + 1 ( − 1 ) n ) f o r d e n o m i n a t o r n = e v e n s + 50 × ( sin ( π n ) + 1 ( − 1 ) n ) f o r d e n o m i n a t o r n = o d d s = ( 50 × 1 ) + ( 50 × − 1 ) = 50 − 50 = 0 f(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 ) = 50 × ( ( − 1 ) n s i n ( πn ) + 1 ) f or d e n o mina t or n = e v e n s + 50 × ( ( − 1 ) n s i n ( πn ) + 1 ) f or d e n o mina t or n = o dd s = ( 50 × 1 ) + ( 50 × − 1 ) = 50 − 50 = 0
∴ f ( n ) = ∑ n = 1 100 sin ( π n ) + 1 ( − 1 ) n = 0 \therefore \ f(n) = \sum_{n = 1}^{100}\frac{\sin(\pi n)+1}{(-1)^n} = 0 ∴ f ( n ) = n = 1 ∑ 100 ( − 1 ) n sin ( πn ) + 1 = 0 Also, for:
f ( m ) = ∏ m = 1 1000 1 cos ( π m ) 2 f(m) = \sqrt{\prod_{m=1}^{1000}\frac{1}{\cos (\pi m)^{2}}} f ( m ) = m = 1 ∏ 1000 cos ( πm ) 2 1 f o r a l l m ∈ N , 1 cos ( π m ) 2 = 1 for \ all \ {m \in \mathbb{N}}, \frac{1} {\cos(\pi m)^2} = 1 f or a ll m ∈ N , c o s ( πm ) 2 1 = 1
Therefore the product of:
∏ m = 1 1000 1 cos ( π m ) 2 = 1 × 1 × 1 × . . . . . . . . = 1 ∴ t h e s q u a r e r o o t o f t h e p r o d u c t o f : ∏ m = 1 1000 1 cos ( π m ) 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 m = 1 ∏ 1000 cos ( πm ) 2 1 = 1 × 1 × 1 × ........ = 1 ∴ t h e s q u a re roo t o f t h e p ro d u c t o f : m = 1 ∏ 1000 cos ( πm ) 2 1 = 1 = 1
∴ f ( m ) = ∏ m = 1 1000 1 cos ( π m ) 2 = 1 \therefore \ f(m) = \sqrt{\prod_{m=1}^{1000}\frac{1}{\cos (\pi m)^{2}}} = 1 ∴ f ( m ) = m = 1 ∏ 1000 cos ( πm ) 2 1 = 1
Therefore, the final answer to the expression is given by:
∴ ( log 10 ( 100 0 100 ) 100 + ∑ n = 1 100 sin ( π n ) + 1 ( − 1 ) n ) . ∏ m = 1 1000 1 cos ( π m ) 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}}} ∴ ( 100 log 10 ( 100 0 100 ) + n = 1 ∑ 100 ( − 1 ) n sin ( πn ) + 1 ) . m = 1 ∏ 1000 cos ( πm ) 2 1
= ( 3 + 0 ) × 1 = (3 + 0) \times 1 = ( 3 + 0 ) × 1 = 3 × 1 = 3 \times 1 = 3 × 1 = 3 = 3 = 3
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