Question

已知數(shù)列{a_n}滿足a_n=

1

(n+1) n +n n+1

，求數(shù)列{a_n}前n項(xiàng)和．

Answer 1

考點(diǎn)：數(shù)列的求和

專題：等差數(shù)列與等比數(shù)列

分析：把數(shù)列的通項(xiàng)進(jìn)行分母有理化，列項(xiàng)后利用裂項(xiàng)相消法求數(shù)列{a_n}前n項(xiàng)和．

解答： 解：∵a_n=

1

(n+1) n +n n+1

=

(n+1) n -n n+1

n(n+1)

=

1

n

-

1

n+1

，
∴Sn=(1-

1

2

)+(

1

2

-

1

3

)+…+(

1

n

-

1

n+1

)=1-

1

n+1

=

n+1 -1

n+1

．

點(diǎn)評(píng)：本題考查了數(shù)列的求和，考查了裂項(xiàng)相消法，關(guān)鍵是分母有理化后的列項(xiàng)，是中檔題．