斐波那契数列通项公式的推导
发布时间：2020-05-08 02:37:13
斐波那契数列通项公式的推导
斐波那契数列：1，1，2，3，5，8，13，21……
如果设F(n)为该数列的第n项(n∈N+).
则![](https://wkrtcs.bdimg.com/rtcs/image?w=312.8&md5sum=390fa99d038db42975e6022ee5fd216e&sign=ac50bc6026&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=5&ipr={"t":"img","w":"312.8","h":"22.666666666667","dataType":"gif","c":"word/media/image1.png"})
显然这是一个线性递推数列.
推导方法一：利用特征方程
线性递推数列的特征方程为：![](https://wkrtcs.bdimg.com/rtcs/image?w=66.866666666667&md5sum=390fa99d038db42975e6022ee5fd216e&sign=ac50bc6026&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=5&ipr={"t":"img","w":"66.866666666667","h":"20.466666666667","dataType":"gif","c":"word/media/image2.png"})
解得![](https://wkrtcs.bdimg.com/rtcs/image?w=161.8&md5sum=390fa99d038db42975e6022ee5fd216e&sign=ac50bc6026&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=5&ipr={"t":"img","w":"161.8","h":"45.333333333333","dataType":"gif","c":"word/media/image3.png"})
.
则![](https://wkrtcs.bdimg.com/rtcs/image?w=124.06666666667&md5sum=390fa99d038db42975e6022ee5fd216e&sign=ac50bc6026&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=5&ipr={"t":"img","w":"124.06666666667","h":"23.733333333333","dataType":"gif","c":"word/media/image4.png"})
.
![](https://wkrtcs.bdimg.com/rtcs/image?w=199.53333333333&md5sum=390fa99d038db42975e6022ee5fd216e&sign=ac50bc6026&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=5&ipr={"t":"img","w":"199.53333333333","h":"48.533333333333","dataType":"gif","c":"word/media/image5.png"})
解得![](https://wkrtcs.bdimg.com/rtcs/image?w=129.46666666667&md5sum=390fa99d038db42975e6022ee5fd216e&sign=ac50bc6026&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=5&ipr={"t":"img","w":"129.46666666667","h":"44.2","dataType":"gif","c":"word/media/image6.png"})
![](https://wkrtcs.bdimg.com/rtcs/image?w=249.2&md5sum=390fa99d038db42975e6022ee5fd216e&sign=ac50bc6026&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=5&ipr={"t":"img","w":"249.2","h":"61.466666666667","dataType":"gif","c":"word/media/image7.png"})
推导方法二：待定系数法
设常数![](https://wkrtcs.bdimg.com/rtcs/image?w=23.733333333333&md5sum=390fa99d038db42975e6022ee5fd216e&sign=ac50bc6026&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=5&ipr={"t":"img","w":"23.733333333333","h":"20.466666666667","dataType":"gif","c":"word/media/image8.png"})
，使得.
则![](https://wkrtcs.bdimg.com/rtcs/image?w=107.86666666667&md5sum=390fa99d038db42975e6022ee5fd216e&sign=ac50bc6026&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=5&ipr={"t":"img","w":"107.86666666667","h":"20.466666666667","dataType":"gif","c":"word/media/image10.png"})
n≥3时，有
![](https://wkrtcs.bdimg.com/rtcs/image?w=312.8&md5sum=390fa99d038db42975e6022ee5fd216e&sign=ac50bc6026&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=5&ipr={"t":"img","w":"312.8","h":"118.66666666667","dataType":"gif","c":"word/media/image11.png"})
将以上n-2个式子相乘，得：
![](https://wkrtcs.bdimg.com/rtcs/image?w=257.8&md5sum=390fa99d038db42975e6022ee5fd216e&sign=ac50bc6026&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=5&ipr={"t":"img","w":"257.8","h":"23.733333333333","dataType":"gif","c":"word/media/image12.png"})
![](https://wkrtcs.bdimg.com/rtcs/image?w=165.06666666667&md5sum=390fa99d038db42975e6022ee5fd216e&sign=ac50bc6026&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=5&ipr={"t":"img","w":"165.06666666667","h":"22.666666666667","dataType":"gif","c":"word/media/image13.png"})
上式可化简为：![](https://wkrtcs.bdimg.com/rtcs/image?w=158.53333333333&md5sum=390fa99d038db42975e6022ee5fd216e&sign=ac50bc6026&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=5&ipr={"t":"img","w":"158.53333333333","h":"23.733333333333","dataType":"gif","c":"word/media/image14.png"})
![](https://wkrtcs.bdimg.com/rtcs/image?w=312.8&md5sum=390fa99d038db42975e6022ee5fd216e&sign=ac50bc6026&rtcs_flag=1&rtcs_ver=3.1&l=webapp&bucketNum=5&ipr={"t":"img","w":"312.8","h":"295.53333333333","dataType":"gif","c":"word/media/image15.png"})
的一解为
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