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AIME 2001 II · 第 3 题

AIME 2001 II — Problem 3

专题
Contest Math
难度
L4
来源
AIME

题目详情

Problem

Given that

x1=211,x2=375,x3=420,x4=523, andxn=xn−1−xn−2+xn−3−xn−4 when n≥5,\begin{aligned}x_{1}&=211,\\ x_{2}&=375,\\ x_{3}&=420,\\ x_{4}&=523,\ \text{and}\\ x_{n}&=x_{n-1}-x_{n-2}+x_{n-3}-x_{n-4}\ \text{when}\ n\geq5, \end{aligned} find the value of x531+x753+x975x_{531}+x_{753}+x_{975}.

解析

Solution

We find that x5=267x_5 = 267 by the recursive formula. Summing the recursions

xn=xn−1−xn−2+xn−3−xn−4xn−1=xn−2−xn−3+xn−4−xn−5\begin{aligned} x_{n}&=x_{n-1}-x_{n-2}+x_{n-3}-x_{n-4} \\ x_{n-1}&=x_{n-2}-x_{n-3}+x_{n-4}-x_{n-5} \end{aligned} yields xn=−xn−5x_{n} = -x_{n-5}. Thus xn=(−1)kxn−5kx_n = (-1)^k x_{n-5k}. Since 531=106⋅5+1, 753=150⋅5+3, 975=194⋅5+5531 = 106 \cdot 5 + 1,\ 753 = 150 \cdot 5 + 3,\ 975 = 194 \cdot 5 + 5, it follows that

x531+x753+x975=(−1)106x1+(−1)150x3+(−1)194x5=211+420+267=898.x_{531} + x_{753} + x_{975} = (-1)^{106} x_1 + (-1)^{150} x_3 + (-1)^{194} x_5 = 211 + 420 + 267 = \boxed{898}.

Solution Variant

The recursive formula suggests telescoping. Indeed, if we add xnx_n and xn−1x_{n-1}, we have xn+xn−1=(xn−1−xn−2+xn−3−xn−4)+(xn−2−xn−3+xn−4−xn−5)=xn−1−xn−5x_n + x_{n-1} = (x_{n-1} - x_{n-2} + x_{n-3} - x_{n-4}) + (x_{n-2} - x_{n-3} + x_{n-4} - x_{n-5}) = x_{n-1} - x_{n-5}.

Subtracting xn−1x_{n-1} yields xn=−xn−5  ⟹  xn=−(−(xn−10))=xn−10x_n = -x_{n-5} \implies x_n = -(-(x_{n-10})) = x_{n-10}.

Thus,

x531+x753+x975=x1+x3+x5=x1+x3+(x4−x3+x2−x1)=x2+x4=375+523=898.x_{531} + x_{753} + x_{975} = x_1 + x_3 + x_5 = x_1 + x_3 + (x_4 - x_3 + x_2 - x_1) = x_2 + x_4 = 375 + 523 = \boxed{898}. Notice that we didn't need to use the values of x1x_1 or x3x_3 at all.

Non-Rigorous Solution

Calculate the first few terms:

211,375,420,523,267,−211,−375,−420,−523,…211,375,420,523,267,-211,-375,-420,-523,\dots At this point it is pretty clear that the sequence is periodic with period 10 (one may prove it quite easily like in solution 1) so our answer is obviously 211+420+267=898211+420+267=\boxed{898}

~Dhillonr25

Video Solution by OmegaLearn

https://youtu.be/lH-0ul1hwKw?t=870

~ pi_is_3.14