dgn induksi matematika, buktikan kebenaran rumus berikut berlaku untuk semua n bilangan asli! 3+5+7+9+...+(2n-1) = n(n+2)

jawab

3+5+7+9+...+(2n+1) = n(n+2)

(i) n = 1
2n +1 = n(n+2)
3 = 3 

(ii) n = k
3+5+7+....+2k+1 = k(k+2)

(iii) n = k + 1
{3 + 5 + 7  + ...+(2k+1)} +  {2(k+1) + 1} =  (k+1)(k +1 + 2)
k(k+2)  + (2k + 3) = (k+1)(k+3)
k²+ 2k + 2k + 3  = (k +1)(k+3)
k²+4k+ 3 = (k+1)(k+3)
(k+1)(k+3) = (k+1)(k+3)