Fun With Sequences 2

can you please elaborate it ....couldn't get the whole idea of it

Definitely. So, the main idea of it is not entirely obvious, but you have to look for it to solve it. If you notice, the digits used in each number in each term range from 1-3. This seems a little unusual, because usually they could range from 0-9. You have to think a little out of the box to figure out the numbers are in base 4 . Now, to elaborate on what that means. Mostly we use base 10, where the digits used are 0-9 and for each digit $a_n$ going to the left in a number, that digit represents $a \cdot 10 ^ n$ , starting with $n = 0$ in the ones digit, and $n = n + 1$ with each digit. So, in base 4, each digit $a_n$ , starting with $n = 0$ at the ones and going left, represents $a \cdot 4 ^ n$ . Now, you can convert the base 4 numbers to base 10 by adding powers of 4. The third term, 11, is equivalent to $1 \cdot 4 ^ 0 + 1 \cdot 4 ^ 1$ , or 5. Doing this for each number gives you a sequence of the first primes in base 10. So now you just find the 8th prime and split it into powers of 4. 19 equals $1 \cdot 4 ^ 2 + 0 \cdot 4 ^ 1 + 3 \cdot 4 ^ 0$ . The answer is $\boxed{103}$