What is the value of $x_{100}$

Question

Let $x_1,x_2,x_3,\ldots$ list the numbers that can be written as a sum of one or more distinct powers of 3, increasing order $(x_1<x_2<x_3<\cdots)$ For example $x_1=3^0=1 ; x_2= 3^1 =3 ; x_3 =3^1+3^0 = 4$. What is the value of $x_{100}$?

I tried to create a geometric series with the powers of 3 but the number I get as a result is too big to compute. Can someone share some light as to how to solve this?

Answer 1

Since $100 = \overline{1100100}_2$ we have $$ x_{100} = 0\cdot 3^0 +0\cdot 3^1+ 1\cdot 3^2+0\cdot 3^3+ 0\cdot 3^4+1\cdot 3^5+1\cdot 3^6 =981$$

Answer 2

The numbers in the series are precisely those whose ternary representation contains no $2$s (I.e., contains only $1$s and $0$s). These can be put into order-preserving correspondence with all binary numbers by taking the representation and re-interpreting it as binary.

So write $100$ in binary, then re-interpret that representation as ternary.