设数列{xn}各项为正,且满足x1²+x2²+x3²+……+xn²=2n²+2n我求出xn=2√n,如何证明x1x2+x2x3+x3x4……+xnx(n+1)
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![设数列{xn}各项为正,且满足x1²+x2²+x3²+……+xn²=2n²+2n我求出xn=2√n,如何证明x1x2+x2x3+x3x4……+xnx(n+1)](/uploads/image/z/6072052-4-2.jpg?t=%E8%AE%BE%E6%95%B0%E5%88%97%7Bxn%7D%E5%90%84%E9%A1%B9%E4%B8%BA%E6%AD%A3%2C%E4%B8%94%E6%BB%A1%E8%B6%B3x1%26%23178%3B%2Bx2%26%23178%3B%2Bx3%26%23178%3B%2B%E2%80%A6%E2%80%A6%2Bxn%26%23178%3B%3D2n%26%23178%3B%2B2n%E6%88%91%E6%B1%82%E5%87%BAxn%3D2%E2%88%9An%2C%E5%A6%82%E4%BD%95%E8%AF%81%E6%98%8Ex1x2%2Bx2x3%2Bx3x4%E2%80%A6%E2%80%A6%2Bxnx%28n%2B1%29)
设数列{xn}各项为正,且满足x1²+x2²+x3²+……+xn²=2n²+2n我求出xn=2√n,如何证明x1x2+x2x3+x3x4……+xnx(n+1)
设数列{xn}各项为正,且满足x1²+x2²+x3²+……+xn²=2n²+2n
我求出xn=2√n,如何证明x1x2+x2x3+x3x4……+xnx(n+1)
设数列{xn}各项为正,且满足x1²+x2²+x3²+……+xn²=2n²+2n我求出xn=2√n,如何证明x1x2+x2x3+x3x4……+xnx(n+1)
(2 )∵
1xn+xn+1
=
12 (n+n+1)
=
12
(
n+1
−
n
)
∴
1x1+x 2
+
1x2+x3
+…+
1xn+xn+1
=
12
(
n+1
−
1
)=3
∴n=48
(3)xnxn+1=2
n
2
n+1
=4
nn+1
<4
n+(n+1)2
=4n+2
∴x1x2+x2x3+…xnxn+1<(4×1+2)+(4×2+2)+…(4n+2)=
6+(4n+2)2
n=2[(n+1)2-1].