2016 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 6

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 $\fbox{9}$ -andliu766

2016 UNM-PNM Contest II (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
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2016 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 6

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