數學歸納法.................................

設 P(n) 為陳述句:
"5n (4n-1)+1 能被16整除"
當 n = 1 時:
5n (4n-1)+1 = 5 × 3 + 1 = 16
所以 P(1) 為正確.
假設 P(k) 為正確, 即 5k (4k-1)+1 = 16M, 其中 M 為某自然數.
當 n = k + 1 時:
[5k+1 (4k+3)+1] - [5k (4k-1)+1] = 5k+1 (4k+3) - 5k (4k-1)
[5k+1 (4k+3)+1] - 16M = 5k [5(4k+3) - (4k-1)]
[5k+1 (4k+3)+1] - 16M = 5k [16k + 16]
5k+1 (4k+3)+1 = 16M + 16(k + 1) × 5k
所以 5k+1 (4k+3)+1 可被 16 整除, 即 P(k + 1) 亦是正確.
所以 P(n) 對所有正整數都是正確的.