Difference between revisions of "2006 AIME II Problems/Problem 11"

Revision as of 12:40, 19 October 2019

Problem

A sequence is defined as follows $a_1=a_2=a_3=1,$ and, for all positive integers $n, a_{n+3}=a_{n+2}+a_{n+1}+a_n.$ Given that $a_{28}=6090307, a_{29}=11201821,$ and $a_{30}=20603361,$ find the remainder when $\sum^{28}_{k=1} a_k$ is divided by 1000.

Solution

Solution 1

Define the sum as $s$ . Since $a_n\ = a_{n + 3} - a_{n + 2} - a_{n + 1}$ , the sum will be:

$s = a_{28} + \sum^{27}_{k=1} (a_{k+3}-a_{k+2}-a_{k+1}) \\ s = a_{28} + \left(\sum^{30}_{k=4} a_{k} - \sum^{29}_{k=3} a_{k}\right) - \left(\sum^{28}_{k=2} a_{k}\right)\\ s = a_{28} + (a_{30} - a_{3}) - \left(\sum^{28}_{k=2} a_{k}\right) = a_{28} + a_{30} - a_{3} - (s - a_{1})\\ s = -s + a_{28} + a_{30}$

Thus $s = \frac{a_{28} + a_{30}}{2}$ , and $a_{28},\,a_{30}$ are both given; the last four digits of their sum is $3668$ , and half of that is $1834$ . Therefore, the answer is $\boxed{834}$ .

Solution 2

Since the problem only asks for the first 28 terms and we only need to calculate mod 1000, we simply bash the first 28 terms:

$a_{1}\equiv 1 \pmod 1000$ $a_{2}\equiv 1 \pmod 1000$ $a_{3}\equiv 1 \pmod 1000$ $a_{4}\equiv 3 \pmod 1000$ $a_{5}\equiv 5 \pmod 1000$ $\cdots$ $a_{25} \equiv 793 \pmod 1000$ $a_{26} \equiv 281 \pmod 1000$ $a_{27} \equiv 233 \pmod 1000$ $a_{28} \equiv 307 \pmod 1000$

Adding all the residues shows the sum is congruent to $\boxed{834}$ mod 1000.

~ I-_-I

Solution 3 (some guessing involved)

All terms in the sequence are sums of previous terms, so the sum of all terms up to a certain point must be some linear combination of the first three terms. Also, we are given $a_{28}, a_{29},$ and $a_{30}$ , so we can guess that there is some way to use them in a formula. Namely, we guess that there exists some $p, q, r$ such that $\sum_{k=1}^{n}{a_k} = pa_n+qa_{n+1}+ra_{n+2}$ . From here, we list out the first few terms of the sequence and the cumulative sums, and with a little bit of substitution and algebra we see that $(p, q, r) = (\frac{1}{2}, 0, \frac{1}{2})$ , at least for the first few terms. From this, we have that $\sum_{k=1}^{28}{a_k} = \frac{a_{28}+a_{30}}{2} \equiv{\boxed{834}}(\mod 1000)$ .

Solution by zeroman

@@ Line 15: / Line 15: @@
 === Solution 2 ===
-Very bad solution: Brute Force. Since the problem asks for the answer of the end value when divided by 1000, it wouldn't be that difficult because you only need to keep track of the last 3 digits.
+Since the problem only asks for the first 28 terms and we only need to calculate mod 1000, we simply bash the first 28 terms:
+<math>a_{1}\equiv 1 \pmod 1000</math>
+<math>a_{2}\equiv 1 \pmod 1000</math>
+<math>a_{3}\equiv 1 \pmod 1000</math>
+<math>a_{4}\equiv 3 \pmod 1000</math>
+<math>a_{5}\equiv 5 \pmod 1000</math>
+<math>\cdots</math>
+<math>a_{25} \equiv 793 \pmod 1000</math>
+<math>a_{26} \equiv 281 \pmod 1000</math>
+<math>a_{27} \equiv 233 \pmod 1000</math>
+<math>a_{28} \equiv 307 \pmod 1000</math>
+Adding all the residues shows the sum is congruent to <math>\boxed{834}</math> mod 1000.
+~ I-_-I
 === Solution 3 (some guessing involved) ===

2006 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 10	Followed by Problem 12
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