Question

數學歸納法證明（n+1）+（n+2）+…+（n+n）=

n(3n+1)

2

的第二步中，當n=k+1時等式左邊與n=k時等式左邊的差等于______．

Answer 1

由題意，n=k時，則（k+1）+（k+2）+…+（k+k）=

k(3k+1)

2

當n=k+1時，左邊=（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
∴當n=k+1時等式左邊與n=k時等式左邊的差等于3k+2
故答案為 3k+2