题干

已知点列An(xn,0),n∈N*,其中x1=0,x2=1.A3是线段A1A2的中点,A4是线段A2A3的中点,…,An+2是线段AnAn+1的中点,…设an=xn+1﹣xn.

(Ⅰ)写出xn与xn﹣1、xn﹣2(n≥3)之间的关系式并计算a1,a2,a3;

(Ⅱ)猜想数列{an}的通项公式,并用数学归纳法加以证明.

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答案(点此获取答案解析)

解:(Ⅰ) xn=xn−1+xn−2

同类题1

通读下面短文,掌握其大意,然后在各题所给的四个选项中选出一个最佳答案。

    It was a lovely spring afternoon. My classmates and I were playing happily on the playground when I let out, “Ow! Ow! Something in my shoe is biting me.”

    Everyone was shocked by the cry. They took me into a classroom and were about to take off my 1. “Which foot is it?” one asked. “Let us have a look.”

    Suddenly, I remembered the holes in 2 socks. My family was very poor during those years. I wore welfare socks, which cost only a little, but those 3 welfare socks didn't last long. They soon had 4 at the bottom.

    I refused to take off my shoes. I 5 stand others seeing the holes in my sock. I tried to hold back my tears. Yet, each time the thing 6 my shoe bit me, tears raced down my face.

    My teacher, Miss Diane, hurried into the 7. “What's wrong?” she asked.

    “Something is biting her right foot, 8 she doesn't let us take off her shoe.” one of my classmates answered.

    Miss Diane lived next door to me. She 9 everything about my family. She put both hands on my shaking shoulders and 10 into my painful and hopeless eyes.

    “Oh, yes, it must be a sock-eating ant,” she said, as if she had 11 seen the thing inside the shoe. “I had a bite from one of those ants. By the time I got my shoe off, it had 12 almost the whole bottom off my sock.” My classmates nodded while they were listening to the teacher 13, although they all looked a little puzzled.

    Miss Diane took off my right shoe and sock and shook them over the dustbin. Two red ants14it.

    “Just what I thought. The ants have eaten part of her sock.” When she stroked an alcohol cotton ball on the bites, she added,” You are such a 15 girl to take so many bites.”

    The alcohol felt cool on the bites and a little girl's pride was saved by the “sock-eating ant” story.