Answer :
Answer:
-2007/4018
Step-by-step explanation:
1. Given
(1/3+1/4+...+1/2009)(1+1/2+...1/2008) - (1+1/3+1/4+...+1/2009)(1/2+1/3+...+1/2008)
2. Set new variable
k = 1/3+1/4+...+1/2008
3. Rewrite the equation
(k+1/2009)(1+1/2+k) - (1+k+1/2009)(1/2+k)
4. Simplify
(k+1/2009)(k+3/2) - (k+2010/2009)(k+1/2)
5. Expand
[k^2 + (1/2009 + 3/2)k + (1/2009)(3/2)] - [k^2 - (2010/2009+1/2)k - (2010/2009)(1/2)]
6. Rewrite
(1/2009 + 3/2 - 2010/2009 - 1/2)k + (1/2009)(3/2) - (2010/2009)(1/2)
7. Simplify
(3/2 - 2009/2009 - 1/2)k + 1/2(3/2009-2010/2009)
8. Evaluate
0*k + 1/2(-2007/2009)
= -2007/4018
The solution of the expression (1/3+1/4+...+1/2009)(1+1/2+...+1/2008)-(1+1/3+1/4+...+1/2009)(1/2+1/3+...+1/2008) is -2007/4018.
What is expression?
An expression is a combination of numbers, symbols or variables that have coefficients.
How to evaluate expression?
We have to evaluate the expression:
(1/3+1/4+...+1/2009)(1+1/2+...1/2008) - (1+1/3+1/4+...+1/2009)(1/2+1/3+...+1/2008)
Set new variable , k = 1/3+1/4+...+1/2008
Rewrite the equation
(k+1/2009)(1+1/2+k) - (1+k+1/2009)(1/2+k)
Simplify the equation:
(k+1/2009)(k+3/2) - (k+2010/2009)(k+1/2)
Expand
[k^2 + (1/2009 + 3/2)k + (1/2009)(3/2)] - [k^2 - (2010/2009+1/2)k - (2010/2009)(1/2)]
Rewrite
(1/2009 + 3/2 - 2010/2009 - 1/2)k + (1/2009)(3/2) - (2010/2009)(1/2)
Simplify
(3/2 - 2009/2009 - 1/2)k + 1/2(3/2009-2010/2009)
Evaluate
0*k + 1/2(-2007/2009)
= -2007/4018
Hence the solution of expression is -2007/4018.
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