2007 AIME II Problems/Problem 12

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Problem

The increasing geometric sequence $x_{0},x_{1},x_{2},\ldots$ consists entirely of integral powers of $3.$ Given that

$\sum_{n=0}^{7}\log_{3}(x_{n}) = 308$ and $56 \leq \log_{3}\left ( \sum_{n=0}^{7}x_{n}\right ) \leq 57,$

find $\displaystyle \log_{3}(x_{14}).$

Solution

Suppose that $\displaystyle x_0 = a$ , and that the common ratio between the terms is $r$ .

The first conditions tells us that $\displaystyle \log_3 a + \log_3 ar \ldots \log_3 ar^7 = 308$ . Using the rules of logarithms, we can simplify that to $\displaystyle \log_3 a^8r^{1 + 2 + \ldots + 7} = 308$ . Thus, $\displaystyle a^8r^{28} = 3^{308}$ . Since all of the terms of the geometric sequence are integral powers of $3$ , we know that both $a$ and $r$ must be powers of 3. Denote $\displaystyle 3^x = a$ and $\displaystyle 3^y = r$ . We find that $8x + 28y = 308$ . The possible positive integral pairs of $(x,y)$ are $(35,1),\ (28,3),\ (21,5),\ (14,7),\ (7,9),\ (0,11)$ .

The second condition tells us that $56 \le \log_3 (a + ar + \ldots ar^7) \le 57$ . Using the sum formula for a geometric series and substituting $x$ and $y$ , this simplifies to $3^{56} \le 3^x \frac{3^{8y} - 1}{3^y-1} \le 3^{57}$ . The fractional part $\approx \frac{3^{8y}}{3^y} = 3^{7y}$ . Thus, we need $\approx 56 \le x + 7y \le 57$ . Checking the pairs above, only $\displaystyle (21,5)$ is close.

Our solution is therefore $\log_3 (ar^{14}) = \log_3 3^x + \log_3 3^{14y} = x + 14y = 091 \displaystyle$ .

2007 AIME II (Problems • Resources)
Preceded by Problem 11	Followed by Problem 13
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

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2007 AIME II Problems/Problem 12

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