Difference between revisions of "2017 UNCO Math Contest II Problems/Problem 10"

Latest revision as of 13:58, 12 April 2023

Powerless Progressions

Find an infinite sequence of integers $a_1, a_2, a_3, \ldots$ that has all of these properties:

(1) $a_n = c + dn$ with c and d the same for all $n = 1, 2, 3, \ldots$

(2) $c$ and $d$ are positive integers, and

(3) no number in the sequence is the $r^{th}$ power of any integer, for any power $r = 2, 3, 4, \ldots$

Reminder: Justify answers. In particular, for maximum credit, make it clear in your presentation that your sequence possesses the third property.

2, 6, 10, 14… (& other possibilities)

Assume $a_n=x^r.$ We now check modulo 4, seeing if any possible $a_n$ are congruent to 2 mod 4.

If $x$ is 0 mod 4, $x$ is a multiple of 4 and can never become 2 mod 4 when exponentiated.

If $x$ is 1 or 3 mod 4, $x$ is odd and cannot become even when exponentiated.

If $x$ is 2 mod 4, $x^r$ is a multiple of 4 for $r \ge 2,$ which is not equivalent to 2 mod 4.

Therefore, $a_n=2+4n$ can never be an $r^{th}$ power of an integer.

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 == Problem ==
+ Powerless Progressions
+Find an infinite sequence of integers <math>a_1, a_2, a_3, \ldots</math>  that has all of
+these properties:
+(1) <math>a_n = c + dn</math> with c and d the same for all <math>n = 1, 2, 3, \ldots</math>
+(2) <math>c</math> and <math>d</math> are positive integers, and
+(3) no number in the sequence is the <math>r^{th}</math> power of any integer, for any power <math>r = 2, 3, 4, \ldots</math>
+Reminder: Justify answers. In particular, for maximum credit, make it clear in your presentation
+that your sequence possesses the third property.
 == Solution ==
+, 6, 10, 14… (& other possibilities)
+Assume <math>a_n=x^r.</math> We now check modulo 4, seeing if any possible <math>a_n</math> are congruent to 2 mod 4.
+If <math>x</math> is 0 mod 4, <math>x</math> is a multiple of 4 and can never become 2 mod 4 when exponentiated.
+If <math>x</math> is 1 or 3 mod 4, <math>x</math> is odd and cannot become even when exponentiated.
+If <math>x</math> is 2 mod 4, <math>x^r</math> is a multiple of 4 for <math>r \ge 2,</math> which is not equivalent to 2 mod 4.
+Therefore, <math>a_n=2+4n</math> can never be an <math>r^{th}</math> power of an integer.
 == See also ==

2017 UNCO Math Contest II (Problems • Answer Key • Resources)
Preceded by Problem 9	Followed by Problem 11
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10
All UNCO Math Contest Problems and Solutions