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Problem of the Day #244: Sequential Shell Building November 18, 2011

Posted by Albert in : potd , trackback

Arjun is constructing a shell pattern for art as seen below, for some monotonically increasing sequence $\{a_k\}$ with $a_0 = 1$.

The right angles for all triangles are fixed at point $P$. The addition of the last triangle (with leg lengths of $a_{n+3}$ and $a_{n+4}$, $n \ge 1$) increases the shape’s area by a factor of:

$$\frac{8 a_n^2 + 10 a_n a_{n-1} + 3 a_{n-1}^2}{3 a_n^2 + 4 a_n a_{n-1} + a_{n-1}^2}$$

Find $a_{15}$.

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