Question

The increasing sequence $1,$ $3,$ $4,$ $9,$ $10,$ $12,$ $13,$ $\ldots$ consists of all those positive integers which are powers of $3$ or sums of distinct powers of $3.$ Find the $100$ th term of this sequence.

Answer

Answer: 981

Concepts:number base place value

Difficulty rating: 1970

Small Hint:

These are precisely the numbers whose base- $3$ digits are all $0$ or $1$

Big Hint:

Write the index in base $2,$ then reinterpret those same digits in base $3$

Solution:

A sum of distinct powers of $3$ has only $0$ ’s and $1$ ’s in base $3.$ As these strings increase, they occur in the same order as binary numerals with the same digit strings. Since $100=1100100_2,$ the $100$ th positive term is $\begin{aligned} 1100100_3 &=3^6+3^5+3^2\\ &=729+243+9\\ &=981. \end{aligned}$

1986 AIME Problem 7

Answer: 981

Difficulty rating: 1970

Small Hint:

Big Hint:

Solution:

Problem 7 in Other Years