Find the value of the expression

Question

1/1.5+1/5.9+1/9.3+.…….+1/2013.2017

 

a. 2013/2017

b. 504/2017

c. 1008/2017

d. 2016/2017

Answer 1

1, 5, 9, ... is arithmetic with common difference 4.  Each term in the sequence has the form 1+4(n-1) = 4n-3

 

5, 9, 13, ... is arithmetic with common difference 4.  Each term in the sequence has the form 5 + 4(n-1) = 4n + 1

 

Each term in the sum has the form 1/[(4n-3)(4n+1)] (Starting with n = 1 and ending with n = 504)

 

    Using Partial Fraction Decomposition, we have 1 / [(4n-3)(4n+1)] = A/(4n-3) + B/(4n+1)

 

                     So, A(4n+1) + B(4n-3) = 1   

 

                          (4A+4B)n + (A-3B) = 1      4A + 4B = 0 

                                                                    A - 3B = 1    Solving the system, we get A = 1/4, B = -1/4

 

The given sum is equivalent to ∑_{(n=1 to n=504)} [1/[4(4n-3)] - 1/[4(4n+1)]

 

                                          = (1/4)∑_{(n=1 to 504)} [1/(4n-3) - 1/(4n+1)]

 

                                           = (1/4)[(1-1/5) + (1/5-1/9) + (1/9-1/13) + ...+(1/2013-1/2017)]

 

                                           = (1/4)[1 - 1/2017] = 504/2017