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Algebra Difficulty 2.1 Junior Prove it Canada

IMG0 In a sequence with six terms, each term
after the second is the sum of the previous two terms. If the fourth
term is $13$ and the sixth term is $36$ , what is the first term? For some real number $r \neq 0$ , the sequence $5r$ , $5r^2$ , $5r^3$ has the property that the second term plus the third term equals the square of the first term. What is the value of $r$ ? Jimmy wrote four tests last week. The average of his marks on the first, second and third tests was $65$ . The average of his marks on the second, third and fourth tests was $80$ . His mark on the fourth test was $2$ times his mark on the first test.
Determine his mark on the fourth test.

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Solution

Since $y = r(x - 3)(x - r)$ passes through $(0,48)$ , then $48 = r(0-3)(0-r)$ . Thus, $48 = 3r^2$ and so $r^2 = 16$ or $r = \pm 4$ .
With $13\%$ sales tax on an item whose price is $\$B$ , the total cost is $\$(1.13B)$ . With $5\%$ sales tax on an item whose price is $\$B$ , the total cost is $\$(1.05B)$ . From the given information $$(1.13B) -
$(1.05B) = $24 $or$ 1.13B - 1.05B = 24$.

Therefore, $0.08B = 24$ , which gives $B = 300$ . Alternatively, we could note that the difference in total prices is the difference in the amount of tax paid. This is the difference between $13\%$ of the original price and $5\%$ of the original price; this difference is equal to $8\%$ of the original price. If $8\%$ of the original price is equal to $\$24$ , then $1\%$ of the original price is equal to $\$3$ and so the original price is $$3 $\times$ 100 =
$300 $. When$ n=1 $,$ f(2n) = (f(n))^2 $becomes$ f(2) = (f(1))^2 $. Since$ f(1) = 3 $, then$ f(2) = 3^2 = 9 $. When$ m=1 $,$ f(2m+1) = 3f(2m) $becomes$ f(3) = 3f(2) $. Since$ f(2) = 9 $, then$ f(3) = 3 $\cdot$ 9 = 27 $. When$ n=2 $,$ f(2n) = (f(n))^2 $becomes$ f(4) = (f(2))^2 $. Since$ f(2) = 9 $, then$ f(4) = 9^2 = 81 $. Therefore,$ f(2) + f(3) + f(4) = 9 + 27 + 81
= 117$.

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.