Difference between revisions of "1987 AIME Problems/Problem 11"

Revision as of 01:10, 13 May 2020

Problem

Find the largest possible value of $k$ for which $3^{11}$ is expressible as the sum of $k$ consecutive positive integers.

Solutions

Solution 1

Let us write down one such sum, with $m$ terms and first term $n + 1$ :

$3^{11} = (n + 1) + (n + 2) + \ldots + (n + m) = \frac{1}{2} m(2n + m + 1)$ .

Thus $m(2n + m + 1) = 2 \cdot 3^{11}$ so $m$ is a divisor of $2\cdot 3^{11}$ . However, because $n \geq 0$ we have $m^2 < m(m + 1) \leq 2\cdot 3^{11}$ so $m < \sqrt{2\cdot 3^{11}} < 3^6$ . Thus, we are looking for large factors of $2\cdot 3^{11}$ which are less than $3^6$ . The largest such factor is clearly $2\cdot 3^5 = 486$ ; for this value of $m$ we do indeed have the valid expression $3^{11} = 122 + 123 + \ldots + 607$ , for which $k=\boxed{486}$ .

Solution 2

First note that if $k$ is odd, and $n$ is the middle term, the sum is equal to kn. If $k$ is even, then we have the sum equal to $kn+k/2$ which is going to be even. Since $3^{11}$ is odd, we see that $k$ is odd.

Thus, we have $nk=3^{11} \implies n=3^{11}/k$ . Also, note $n-(k+1)/2=0 \implies n=(k+1)/2.$ Subsituting $n=3^{11}/k$ , we have $k^2+k=2*3^{11}$ . Proceed as in solution 1.

@@ Line 1: / Line 1: @@
 == Problem ==
 Find the largest possible value of <math>k</math> for which <math>3^{11}</math> is expressible as the sum of <math>k</math> consecutive [[positive integer]]s.
-== Solution 1==
+==Solutions==
+=== Solution 1===
 Let us write down one such sum, with <math>m</math> terms and first term <math>n + 1</math>:

1987 AIME (Problems • Answer Key • Resources)
Preceded by Problem 10	Followed by Problem 12
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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