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Number Theory Level 2

$1,3,4,9,10,12,13,\ldots$

The above sequence consists of the powers of $3$ or the sum of distinct powers of $3$ , arranged in increasing order.

Find the $100^\text{th}$ term of the sequence.

The answer is 981.

2 solutions

Abdeslem Smahi
Jul 17, 2015

If the terms of the sequence are written in base-3, they comprise the positive integers which do not contain the digit $2$ .

Thus, the terms of the sequence in ascending order are thus:

$1,10,11,100,101,110,111,...$

In the binary scale, these numbers are, of course, $1, 2, 3, ...$ To obtain the 100-th

term of the sequence we just write $100$ in binary $100 = 1100100_2$ and translate

this into ternary: $1100100_3 = 3^6 +3^5 +3^2 = 981.$

Never thought of writing it in binary. I got until the ternary representation, and somehow I messed up, I suppose.

Jake Lai - 5 years, 10 months ago

Me too , i passed a day to think of working with binary and solve it .

Abdeslem Smahi - 5 years, 10 months ago

Very very neat problem!

Jake Lai - 5 years, 10 months ago

Yay, that's exactly how I did it! Upvoted!

Eamon Gupta - 5 years, 10 months ago

3^0=1 ; 3^1=3 ; 3^0+3^1=4; 3^2=9;3^0+3^2=10;3^1+3^2=12;3^0+3^1+3^2=13......... etc i wrote program to execute this pattern and get different solution
why ? :(

Andrew Amin - 4 years, 5 months ago

So did I...On the contrary, I did manually yet got a different answer.

Mohli Thapa - 2 years, 5 months ago

Ryan Chatterjee
Aug 22, 2018

The question becomes a lot simpler when we convert to base 3. Because the powers of 3 must be distinct, they can only appear once or not at all. So their base 3 representation consists only of ones and zeroes, kind of like binary. Writing them in increasing order, it's easy to see that we're just counting in binary. So to find the 100th term in the sequence, convert the number 100 into binary, which gives us 1100100. The 100th term in the sequence is just the number 1100100 in base 3, or 729+243+9=981.

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The answer is 981.

2 solutions

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