Difference between THD equation and design circuit current
Why does the summation of design circuit current equation (Ib) start with 1 while for calculating THD the summation starts with 2?

While the other answers have pointed out that this is from the definition of harmonic distortion, the question suggests broader concern about what the terms mean so I'll try and answer that.
The total current in the circuit is obvious enough.  It is all of the current at all frequencies.  The text assumes &quot;harmonic current&quot;, meaning that all current flows at some fundamental frequency (example:  60 Hz) or a hamonic of it (120 Hz, 180 Hz, ...).  So it adds the fundamental frequency (h=1) to all the harmonics (h=2 ...).
Harmonic distortion is signal that shows up at harmonics of the fundamental.  Total harmonic distortion is the sum of energy in the harmonics divided by the fundamental.  So if you put in 60 Hz and at the output of your system you measure a signal at 120 Hz with 1% amplitude, while your fundamental has been decreased to 99%, you know that your THD is 0.01/0.99 = 0.0101 or about 1%.