题目内容
【题目】已知an=logn+1(n+2)(n∈N+),观察下列运算:a1a2=log23log34= =2;a1a2a3a4a5a6=log23log34…log67lg78=
=3;….定义使a1a2a3…ak为整数的k(k∈N+)叫做希望数,则在区间[1,2016]内所有希望数的和为( )
A.1004
B.2026
C.4072
D.22016﹣2
【答案】B
【解析】解:an=logn+1(n+2)= ,∴a1a2a3…an=
…
=
=k,∴n+2=2k .
n∈[1,2016],∴n=22﹣2,23﹣1,…,210﹣2,
∴在区间[1,2016]内所有希望数的和为=22﹣2+23﹣2+…+210﹣2= ﹣2×9=2026,
故选:B.
an=logn+1(n+2)= ,可得a1a2a3…an=
=k,n=2k﹣2.即可得出.

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