作业帮已知函数f(n)=log(n+1)(n+2)(n∈N*),定义使f(1),f(2)……f(K)为整数的数k(k∈N*)叫企盼数,则在区间[1,100]内这样的企盼数共有______个
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![已知函数f(n)=log(n+1)(n+2)(n∈N*),定义使f(1),f(2)……f(K)为整数的数k(k∈N*)叫企盼数,则在区间[1,100]内这样的企盼数共有______个](/uploads/image/z/11617657-25-7.jpg?t=%E5%B7%B2%E7%9F%A5%E5%87%BD%E6%95%B0f%28n%29%3Dlog%28n%2B1%29%28n%2B2%29%28n%E2%88%88N%2A%29%2C%E5%AE%9A%E4%B9%89%E4%BD%BFf%281%29%2Cf%282%29%E2%80%A6%E2%80%A6f%28K%29%E4%B8%BA%E6%95%B4%E6%95%B0%E7%9A%84%E6%95%B0k%28k%E2%88%88N%2A%29%E5%8F%AB%E4%BC%81%E7%9B%BC%E6%95%B0%2C%E5%88%99%E5%9C%A8%E5%8C%BA%E9%97%B4%5B1%2C100%5D%E5%86%85%E8%BF%99%E6%A0%B7%E7%9A%84%E4%BC%81%E7%9B%BC%E6%95%B0%E5%85%B1%E6%9C%89______%E4%B8%AA)
已知函数f(n)=log(n+1)(n+2)(n∈N*),定义使f(1),f(2)……f(K)为整数的数k(k∈N*)叫企盼数,则在区间[1,100]内这样的企盼数共有______个
已知函数f(n)=log(n+1)(n+2)(n∈N*),定义使f(1),f(2)……f(K)为整数的数k(k∈N*)叫企盼数,则在区间[1,100]内这样的企盼数共有______个
已知函数f(n)=log(n+1)(n+2)(n∈N*),定义使f(1),f(2)……f(K)为整数的数k(k∈N*)叫企盼数,则在区间[1,100]内这样的企盼数共有______个
1＜f(n)=log(n+1)(n+2)≤100
n+1＜(n+1)^f(n)=n+2≤(n+1)^100
n≥1
3＜2^f1≤n+2≤(n+1)^100,
f1≥2
4≤2^f1≤n+2≤(n+1)^100,
n≥2
4≤3^f2≤n+2≤(n+1)^100
f2≥2,
9≤3^f2≤n+2≤(n+1)^100
n≥7
9≤8^f7≤n+2≤(n+1)^100
f7≥2,
64≤8^f7≤n+2≤(n+1)^100
n≥62
64≤63^f62≤n+2≤(n+1)^100
f62≥2,
63^2≤63^f62≤n+2≤(n+1)^100
所以填0.
f(n)=log(n+1)(n+2)=[ln(n+2)]/[ln(n+1)]
a1*a2*a3*......*an
=(ln3/ln2)(ln4/ln3)(ln5/ln4)...[ln(n+1)/ln(n)][ln(n+2)/ln(n+1)]
=1/(ln2)*(ln3/ln3)(ln4/ln4)(ln5/ln5)
....[ln(n)]/[ln(n)]*[...
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f(n)=log(n+1)(n+2)=[ln(n+2)]/[ln(n+1)]
a1*a2*a3*......*an
=(ln3/ln2)(ln4/ln3)(ln5/ln4)...[ln(n+1)/ln(n)][ln(n+2)/ln(n+1)]
=1/(ln2)*(ln3/ln3)(ln4/ln4)(ln5/ln5)
....[ln(n)]/[ln(n)]*[ln(n+1)]/[ln(n+1)]*ln(n+2)
=[ln(n+2)]/ln2
=log(2)(n+2)
=k(k是整数)
得n+2=2^k
n=2^k-2
n<100,得n=2^k-2<100,2^k<102
2^7=128,2^6=64
得k≤6
又n=2^k-2>0,得k>1,即k≥2
即数列{2^k-2},2≤k≤6,
所以k有5个
收起