Harmonious terms

Suppose, to the contrary, that at least three of the statements are true. Then for some distinct indices $a,b,c$ we have $t_a=a, t_b=b, t_c=c$ . Since this sequence is a harmonic sequence, its reciprocals make an arithmetic sequence. The common difference of the arithmetic sequence can be written as follows: $d = \frac{\frac{1}{a}-\frac{1}{b}}{a-b} = \frac{\frac{1}{b}-\frac{1}{c}}{b-c}$ $d= -\frac{1}{ab} = -\frac{1}{bc}$

Therefore $a=c$ . However, this contradicts the assumption that the indices are different. Hence, at most two of the statements can be true.

The infinite list of statements describes the linear function $\ell(n) = n$ . The harmonic progression describes a function $h(n) = \frac a{b+n}$ . The number of true statements is the number of intersection points of the graphs. A line and a hyperbola have at most two intersection points, so that the maximum number is less than or equal to 2.

$h(n)=\frac{k}{a+(n-1)d}$ defines harmonic progression. With h(1)=1 => a=k, h(2)=2 => d=-k/2 we get:

$h(n)=\frac{2}{3-n}$

We realize that the progression has only the 1st and 2nd term of development since h(3) is not defined and negative members are invalid.