2011 UNCO Math Contest II Problems/Problem 7


What is the $\underline{sum}$ of the first $999$ terms of the sequence $1, 2, 3, 6, 7, 14, 15, 30, 31, 62, 63,\cdots$ that appeared on the First Round? Recall that a term in an even numbered position is twice the previous term, while a term in an odd numbered position is one more that the previous term.


Group every even term with the term following it, like so


Since every odd term is 1 less than a power of two, we know each group will be of the form $(2^{n}-2)+(2^{n}-1)$, or just $2^{n+1}-3$. So our sum becomes


Counting yields 500 such groups, leaving us with a geometric series minus $3*500=1500$, giving us




See Also

2011 UNCO Math Contest II (ProblemsAnswer KeyResources)
Preceded by
Problem 6
Followed by
Problem 8
1 2 3 4 5 6 7 8 9 10
All UNCO Math Contest Problems and Solutions