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Difference between revisions of "2016 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 6"

(Created page with "== Problem == For a positive integer <math>k</math> let <math>\sigma(k)</math> be the sum of the digits of <math>k</math>. For example, <math>\sigma(1234) = 1 + 2 + 3 + 4 = 1...")
 
(→‎Solution)
Line 7: Line 7:
  
 
== Solution==
 
== Solution==
−
 
+
We can plug in the function 4 times. We get that
 +
<math>a_2 = 2+0+1+6+2+0+1+6 = 18</math>
 +
<math>a_3 = 1+8 = 9</math>
 +
<math>a_4 = 9</math>
 +
<math>a_5 = 9</math>
 +
So the answer is <math>\fbox{9}</math>
  
 
== See also ==
 
== See also ==

Revision as of 20:01, 6 August 2023

Problem

For a positive integer $k$ let $\sigma(k)$ be the sum of the digits of $k$. For example, $\sigma(1234) = 1 + 2 + 3 + 4 = 10$, while $\sigma(4) = 4$. Let $a_1 = 20162016$ and define $a_{n+1} = \sigma(a_n), n = 1, 2, 3,\cdots$

Find $a_5$.

Solution

We can plug in the function 4 times. We get that $a_2 = 2+0+1+6+2+0+1+6 = 18$ $a_3 = 1+8 = 9$ $a_4 = 9$ $a_5 = 9$ So the answer is $\fbox{9}$

See also

2016 UNM-PNM Contest II (Problems • Answer Key • Resources)
Preceded by
Problem 5 		Followed by
Problem 7
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10
All UNM-PNM Problems and Solutions

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