Question

Evaluate (1/3+1/4+...+1/2009)(1+1/2+...+1/2008)-(1+1/3+1/4+...+1/2009)(1/2+1/3+...+1/2008)

Answer 1

Answer: -2007/4018. hope it helped

Answer 2

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