Question

数学归纳法证明（n+1）+（n+2）+…+（n+n）=

n(3n+1)

2

的第二步中，当n=k+1时等式左边与n=k时等式左边的差等于______．

Answer 1

由题意，n=k时，则（k+1）+（k+2）+…+（k+k）=

k(3k+1)

2

当n=k+1时，左边=（k+1+1）+（k+1+2）+…+（k+1+k-1）+（k+1+k）+（k+1+k+1）
=（k+2）+（k+3）+…+（k+k）+（k+1+k）+（k+1+k+1）
=（k+1）+（k+2）+（k+3）+…+（k+k）+（k+1+k）+（k+1+k+1）-（k+1）
=

k(3k+1)

2

+3k+2
∴当n=k+1时等式左边与n=k时等式左边的差等于3k+2
故答案为 3k+2