## find an item of series

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### find an item of series

mathLover
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Joined: Tue Apr 17, 2012 3:46 am

### Re: find an item of series

Every member of S is of the form $2^x + 2^y + 2^z$, and S can be sorted in increasing order as follows: sort by greater z, then by greater y, and then by greater x. Let us consider the number of elements sorting by greater z:

$2^0 + 2^1 + 2^2$ 1 element for z = 2

$2^0 + 2^1 + 2^3$

$2^0 + 2^2 + 2^3$

$2^1 + 2^2 + 2^3$ 3 elements for z = 3...

${k \choose 2}$ elements for z = k.

Thus, up to z = k, there are a total of ${k+1 \choose 3}$ elements.

The first k such that ${k+1 \choose 3} \geq 100$ is 9, so for the 100th element, z = 9.

Now the 120th element is $2^7 + 2^8 + 2^9$, so we simply work backwards from there:

The 113th element is $2^0 + 2^8 + 2^9$

The 112th element is $2^6 + 2^7 + 2^9$

The 106th element is $2^0 + 2^7 + 2^9$

The 105th element is $2^5 + 2^6 + 2^9$

The 100th element is $2^0 + 2^6 + 2^9$.
icemanfan