Difference between revisions of "2007 AIME II Problems/Problem 12"

Revision as of 20:15, 7 January 2022

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

Solution

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

The first conditions tells us that $\log_3 a + \log_3 ar + \ldots + \log_3 ar^7 = 308$ . Using the rules of logarithms, we can simplify that to $\log_3 a^8r^{1 + 2 + \ldots + 7} = 308$ . Thus, $a^8r^{28} = 3^{308}$ . 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 $3^x = a$ and $3^y = r$ . 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 $(21,5)$ is close.

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

Solution 2

All these integral powers of $3$ are all different, thus in base $3$ the sum of these powers would consist of $1$ s and $0$ s. Thus the largest value $x_7$ must be $3^{56}$ in order to preserve the givens. Then we find by the given that $x_7x_6x_5\dots x_0 = 3^{308}$ , and we know that the exponents of $x_i$ are in an arithmetic sequence. Thus $56+(56-r)+(56-2r)+\dots +(56-7r) = 308$ , and $r = 5$ . Thus $\log_3 (x_{14}) = \boxed{091}$ .

Solution 3

Like above, we use logarithmic identities to convert the problem into workable equations. We begin by labelling the powers of 3. Call $x_0$ , $x_1$ , $x_2$ ..., as $3^n$ , $3^{n+m}$ , and $3^{n+2m}$ ... respectively. With this format we can rewrite the first given equation as $n + n + m + n+2m + n+3m+...+n+7m = 308$ . Simplify to get $2n+7m=77$ . (1)

Now, rewrite the second given equation as $3^{56} \leq \left( \sum_{n=0}^{7}x_{n} \right) \leq 3^{57}$ . Obviously, $x_7$ , aka $3^{n+7m}$ $<3^{57}$ because there are some small fractional change that is left over. This means $n+7m$ is $\leq56$ . Thinking about the geometric sequence, it's clear each consecutive value of $x_i$ will be either a power of three times smaller or larger. In other words, the earliest values of $x_i$ will be negligible compared to the last values of $x_i$ . Even in the best case scenario, where the common ratio is 3, the values left of $x_7$ are not enough to sum to a value greater than 2 times $x_7$ (amount needed to raise the power of 3 by 1). This confirms that $3^{n+7m} = 3^{56}$ . (2)

Use equations 1 and 2 to get $m=5$ and $n=21$ . $\log_{3}{x_{14}} = \log_{3}{3^{21+14*5}} = 21+14*5 = \boxed{091}$

-jackshi2006

@@ Line 24: / Line 24: @@
 Like above, we use logarithmic identities to convert the problem into workable equations. We begin by labelling the powers of 3. Call <math>x_0</math>, <math>x_1</math>, <math>x_2</math>..., as <math>3^n</math>, <math>3^{n+m}</math>, and <math>3^{n+2m}</math>... respectively. With this format we can rewrite the first given equation as <math>n + n + m + n+2m + n+3m+...+n+7m = 308</math>. Simplify to get <math>2n+7m=77</math>. (1)
-Now, rewrite the second given equation as <math>3^{56} \leq ( \sum_{n=0}^{7}x_{n} ) \leq 3^{57}</math>. Obviously, <math>x_7</math>, aka <math>3^{n+7m}</math> <math><3^{57}</math> because there are some small fractional change that is left over. This means <math>n+7m</math> is <math>\leq56</math>. Thinking about the geometric sequence, it's clear each consecutive value of <math>x_i</math> will be either a power of three times smaller or larger. In other words, the earliest values of <math>x_i</math> will be negligible compared to the last values of <math>x_i</math>. Even in the best case scenario, where the common ratio is 3, the values left of <math>x_7</math> are not enough to sum to a value greater than 2 times <math>x_7</math> (amount needed to raise the power of 3 by 1). This confirms that <math>3^{n+7m} = 3^{56}</math>. (2)
+Now, rewrite the second given equation as <math>3^{56} \leq \left( \sum_{n=0}^{7}x_{n} \right) \leq 3^{57}</math>. Obviously, <math>x_7</math>, aka <math>3^{n+7m}</math> <math><3^{57}</math> because there are some small fractional change that is left over. This means <math>n+7m</math> is <math>\leq56</math>. Thinking about the geometric sequence, it's clear each consecutive value of <math>x_i</math> will be either a power of three times smaller or larger. In other words, the earliest values of <math>x_i</math> will be negligible compared to the last values of <math>x_i</math>. Even in the best case scenario, where the common ratio is 3, the values left of <math>x_7</math> are not enough to sum to a value greater than 2 times <math>x_7</math> (amount needed to raise the power of 3 by 1). This confirms that <math>3^{n+7m} = 3^{56}</math>. (2)
-Use equations 1 and 2 to get <math>m=5</math> and <math>n=21</math>. <math>log_{3}{x_{14}} = log_{3}{3^{21+14*5}} = 21+14*5 = \boxed{091}</math>
+Use equations 1 and 2 to get <math>m=5</math> and <math>n=21</math>. <math>\log_{3}{x_{14}} = \log_{3}{3^{21+14*5}} = 21+14*5 = \boxed{091}</math>

2007 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
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