Harmonic Progression - Properties

Finding the terms in the sequence, we see that $$t_3=0$$, $$t_4=-\dfrac{1}{3}$$, $$t_5=0$$
We would notice that all the odd terms are 0, and we are also asked the sum of only even terms, so we do not need to consider those

$$t_6=-\dfrac{1}{5}$$
We see that the even terms are in an HP: $$-1,\ -\dfrac{1}{3},\ -\dfrac{1}{5},\ -\dfrac{1}{7},\ ...$$

The sum we are asked is the inverse of these terms, that is: -1, -3, -5, -7, up to 1012 terms

The sum of this AP would be $$\dfrac{\left[-\left(2\times\ 1\right)+\left(1012-1\right)\left(-2\right)\right]}{2}\times\ 1012$$
Which is equal to $$-1012\times\ 1012\ =\ -1024144$$

Therefore, Option A is the correct answer. 

If x, y and z are in H.P, then 1/x, 1/y and 1/z are in A.P
So (x+y+z)/x, (x+y+z)/y and (x+y+z)/z are in A.P
So (x+y+z)/x - 2, (x+y+z)/y - 2 and (x+y+z)/z - 2 are in A.P
So (y+z-x)/x, (x+z-y)/y and (x+y-z)/z are in A.P

The corresponding AP is 3, 5, 7 . . . The 105th term for the AP= 3+104*2 = 211. Hence the 105th term in the HP = 1/211.