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Note: I just added a derivation of an explicit formula for the terms as the OP requested. From 3,8,17,32,57, if the term is n, the next term is 2n+k where k = 2, 1, -2, -7. The differences of k are -1, -3, -5. If we assume that the next difference is -7, the next k is -7-7=-14 and the next term is 2*57-14 = 100. To get a formula from this, since the sum ...

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If you want the smallest degree polynomial that gives these numbers (which is only one of the infinitely many possible explanations for them), we can use the calculus of finite differences: $$\fbox{3}\quad 8\quad 17\quad 32\quad 57 \\ \fbox{5}\quad 9\quad 15\quad 25\\ \fbox{4}\quad 6\quad 10\\ \fbox{2} \quad 4\\ \fbox{2}$$ Define \begin{align*} ... 1 Hint: One solution can be obtained by taking differences-of-differences. 2 As Qiaochu Yuan posted, what you're looking for is the least common denominator of the ratios involved. Many thanks to him - the post I previously had here did not deal with ratios whose decimal expansions are infinite. I have corrected this post accordingly. We can find the LCD of a set of ratios writing the ratios as fractions with integral numerators and ... 3 We can look at a regular n-gon of "radius" r, i.e., the convex hull of r times the n-th roots of unity. Connecting the vertices of the polygon to the origin gives you n isosceles triangles of area \frac{r^2}{2}\sin \frac{2\pi}{n}. The total area of the n-gon is thusA = \frac{nr^2}{2} \sin \frac{2\pi}{n}. To calculate the length $l$ of ...

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In a sense there must be patterns to be found in rainbows, because they can be described mathematically and math is all about patterns. However, the mathematics of rainbows is quite complex, and if you are looking for patterns that are simple and elegant, then probably the best example of such to be found in a rainbow is the pattern we see. That is, the ...

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Answer is B according to me. Check the number of divisions of the figures. They appear to be a,a+1,b,b+1,c,c+1. 4 in first, 5 in second, 1 in third, 2 in fourth and so on.

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