Question

已知：an=logn+1(n+2)(n∈N*)，觀察下列運(yùn)算：a1•a2=lo

g

3 2

•lo

g

4 3

=

lg3

lg2

•

lg4

lg3

=2，a1•a2•a3•a4•a5•a6=lo

g

3 2

•lo

g

4 3

•…•lo

g

7 6

•lo

g

8 7

=

lg3

lg2

•

lg4

lg3

•…•

lg7

lg6

•

lg8

lg7

=3則當(dāng)a₁•a₂•…•a_k=2012時(shí)，自然數(shù)k為（　�。�

Answer 1

分析：利用對(duì)數(shù)的運(yùn)算性質(zhì)，化簡(jiǎn)a₁•a₂•…•a_k，即可求得結(jié)論．

解答：解：∵an=logn+1(n+2)(n∈N*)，
∴a₁•a₂•…•a_k=lo

g

3 2

•lo

g

4 3

•…•lo

g

7 6

•lo

g	(k+2) (k+1)

=

lg3

lg2

•

lg4

lg3

•…•

lg7

lg6

•

lg(k+2)

lg(k+1)

=

lg(k+2)

lg2

∵a₁•a₂•…•a_k=2012
∴

lg(k+2)

lg2

=2012
∴k+2=2²⁰¹²
∴k=2²⁰¹²-2
故選C．

點(diǎn)評(píng)：本題考查類(lèi)比推理，考查了對(duì)數(shù)的換底公式及對(duì)數(shù)的運(yùn)算性質(zhì)的應(yīng)用，屬于中檔題．