$Odd \ love!^{♥}$

Sum of odd natural numbers till the ${n}^{th}$ term is ${n}^{2}$
Number of terms till 1729 is $\frac{1729+1}{2}=865$
Therefore , $x={865}^{2}$
or, $x=748225$
Sum of digits is $\boxed{28}$

$\begin{matrix} S & = & 1 & + & 3 & + & 5 & + & \dots & + & 1729 \\ S & = & 1729 & + & 1727 & + & 1725 & + & \dots & + & 1 \\ 2S & = & 1730 & + & 1730 & + & 1730 & + & \dots & + & 1730 \end{matrix}$

There are 865 terms in the series. Therefore,

$\begin{matrix} 2S & = & 1730\cdot 865 \\ S & = & \frac { 1730\cdot 865 }{ 2 } \\ S & = & \boxed { 748225 } \end{matrix}$

A little long way :

We can see that the numbers are in AP. So we have $a=1$ and $d=2$ .We also know that , total number of terms is , $\frac{1729+1}{2}=865$ .

Now applying summation of AP formula , which is , $S_{865}=\frac{865}{2} (1729+1)=865 . 865 = 748225$ hence sum of digits is equal to $7+4+8+2+2+5 = \boxed{28}$

This is a simple $Arithmetic \ Progression$ .

The number of terms is given by $T_{n} = a + (n - 1)*d$

So, applying it we get that, $1729 = 1 + (n -1)*2$ $→$ $n = 865$

Now, sum of an AP is given by , $S_{n} =$ $\frac{n}{2}$ $* (a + l)$

Thus, the sum of the given AP is $S_{1729} =$ $\frac{865}{2}$ $* (1 + 1729)$

= $\frac{865 * 1730}{2}$ $= 865^{2}$ $= 748225$

Hence the sum of the digits is $7+4+8+2+2+5 =$ $\boxed{\boxed{28}}$

CHEERS!!