1.Buktikan 1/1^2 + 1/2^2 + 1/3^2 + ... + 1/n^2 <= 2 - 1/n , untuk setiap bil.asli n . 2.Buktikan bahwa n^2 + 3 <= 2^n , untuk semua bil.asli n >= 5 Ket: ^ (pang

Induksi Matematika

1/1² +1/(2²) + 1/(3²) + ... +1/(n²) ≤ 2 - 1/(n)
.
n= 1 → 1/(1²) ≤ 2 -1/1 → 1 ≤ 1 (benar)
n= k → 1/(1²) + 1/(2²) + 1/(3²) +...+1/(k²) ≤ (2 - 1/k)
n= k+ 1 → 2 - 1/(k) + 1/(k+1)² ≤  2 - 1/(k+1) ...kalikan k(k+1)²
.
2k(k+1)²  - (k+1)² + k   ≤ 2k(k+1)² - k(k+1)
2k(k+1)² - k²- 2k -1 + k ≤ 2k(k+1)² - k(k+1)
2k(k+1)² - k²- k -1 ≤ 2k(k+1)² - k(k+1)
2k(k+1)²- k(k+1) - 1 ≤ 2k(k+1)² -k(k+1)
jika f(k)=2k(k+1) -k(k+1)
f(k) - 1 ≤ f(k)
f(n) -1 ≤ f(n), n bilangan asli