Question

已知數(shù)列{a_n}滿足a₁=1,a_n+1=,

(1)計(jì)算a₂,a₃,a₄;

(2)猜測a_n的表達(dá)式,并用數(shù)學(xué)歸納法加以證明.

Answer 1

解:a₂==;?

a₃===;?

a₄===.?

猜想a_n=.

證明:當(dāng)n=1時(shí)顯然成立.?

假設(shè)當(dāng)n=k時(shí)成立,即a_n=.?

當(dāng)n=k+1時(shí),a_n+1===,

∴當(dāng)n=k+1時(shí)也成立.?

∴a_n=對任意n∈N^*成立.