Question

Tunjukkan bahwa 1^2 + 2^2 + 3^2 + . . . + n^2 = n (n+1)(2n+1) / 6
dengan n > sama dengan 1

Answer 1

Verified answer

Induksi matematika
Kls XI sma

1. Bukti n=1 benar
n = 1
1^2 = 1/6 (1) (1 + 1)(2.1 + 1)
1 = 1/6 . 1 . 2 . 3
1 = 1 benar

2.jika n=k benar maka akan dibuktikan jga n= k+1 jga benar
n = k
1^2 + 2^2 + ... + k^2 = 1/6 . k (k + 1)(2k + 1)

n = k + 1
1^1 + 2^2 + ... + k^2 + (k + 1)^2 = 1/6 . (k + 1) ((k + 1) + 1) (2(k + 1) + 1)

Akan dibuktikan ruas kiri sama dg kanan

[1/6 . k(k + 1)(2k + 1)] + (k + 1)^2
(k + 1) [(1/6 k(2k + 1)) + (k + 1)]
(k + 1) [(2k^2 + k)/6 + (k + 1)]
(k + 1) [(2k^2 + k + 6k + 6)/6]
(k + 1) (2k^2 + 7k + 6)/6
1/6 (k + 1)(k + 2)(2k + 3)
1/6 (k + 1)(k + 1 + 1)(2k + 2 + 1)
1/6 (k + 1)((k + 1) + 1)(2(k + 1) + 1)