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1998 CEMC Gauss (Grade 7) Problems/Problem 24

Problem

On a large piece of paper, Dana creates a “rectangular spiral” by drawing line segments of lengths, in cm, of 1, 1, 2, 2, 3, 3, 4, 4, ... as shown. Dana’s pen runs out of ink after the total of all the lengths he has drawn is 3000 cm. What is the length of the longest line segment that Dana draws?

$\text{(A)}\ 38 \qquad \text{(B)}\ 39 \qquad \text{(C)}\ 54 \qquad \text{(D)}\ 55 \qquad \text{(E)}\  50$

Solution

We are finding the sum of the integers from $1$ to $n$ twice, which is $2 \cdot \frac{(n)(n+1)}{2} =  (n)(n+1).$ We need to solve the equation $n(n+1) \le 3000,$ which results in $n = 54.$

The answer is $\text{(C)}.$

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