1+3+6+10+...+ n(n+1) / 2 = n(n+1)(n+2) / 6

Matematika Wajib
Induksi Matematika XI SMA

Soal : 
Buktikan dengan Induksi Matematika 
1 + 3 + 6 + ... + [tex] \frac{n(n+1)}{2} = \frac{n(n+1)(n+2)}{6} [/tex]

Jawab :
Untuk n = 1, maka :
[tex] \frac{1(1 + 1)}{2} = \frac{1(1+1)(1+2)}{6}\\ \frac{1(2)}{2} = \frac{1(2)(3)}{6}\\1 = 1\\TERBUKTI [/tex]

Untuk n = k, maka :
[tex]1 + 3 + 6 + 10 + ... + \frac{k(k + 1)}{2} = \frac{k(k+1)(k+2)}{6}\\DIANGGAP\,\ BENAR[/tex]

Untuk n = k + 1, maka :
[tex]1 + 3 + ... + \frac{k(k+1)}{2} + \frac{(k+1)(k+1+1)}{2} = \frac{(k+1)(k+1+1)(k + 1+2)}{6}\\ \frac{k(k + 1)(k+2)}{6} + \frac{(k+1)(k+2)}{2}= \frac{(k + 1)(k + 2)(k + 3)}{6} \\ \frac{k(k+1)(k+2)}{6}+ \frac{3(k+1)(k + 2)}{6} = \frac{(k+1)(k + 2)(k + 3)}{6}\\ (k+2) [\frac{k(k + 1) + 3(k + 1)}{6}] = \frac{(k+1)(k + 2)(k + 3)}{6}\\ (k + 2) [ \frac{k^2 + k + 3k + 3}{6}] = \frac{(k+1)(k + 2)(k + 3)}{6}\\( k + 2) [ \frac{k^2 + 4k + 3}{6}] = \frac{(k+1)(k + 2)(k + 3)}{6}[/tex]
[tex](k + 2)[ \frac{(k + 1)(k + 3)}{6}] = \frac{(k+1)(k + 2)(k + 3)}{6}\\ \frac{(k + 1)(k + 2)(k + 3)}{6} = \frac{(k+1)(k + 2)(k + 3)}{6}\\\\TERBUKTI... [/tex]