Práctica 1: Inducción (Algoritmos III)

Ejercicio 01.01:

a) P(n) = Σ_i=1..n i = n(n+1)/2

• CB: n = 1

Σ_i=1..1 i = 1

1(1+1)/2 = 1 OK

• PI: P(n)=>P(n+1)

Σ_i=1..n+1 i = Σ_i=1..n i + n+1 = (HI) n(n+1)/2 + 2(n+1)/2 = (n+2)(n+1)/2 OK

b) P(n) = Σ_i=0..n (2*i+1) = (n+1)²

• CB: n = 0

Σ_i=0..0 (2*i+1) = 1 (0+1)² = 1 OK

• PI: P(n)=>P(n+1)

Σ_i=0..n+1 (2*i+1) = Σ_i=0..n 2*i+1 + 2*(n+1)+1 = (HI) (n+1)² + 2n + 3 = n² + 2*n + 1 + 2n + 3 = n² + 4*n + 4 = (n+2)² OK
c)
d)
e)
f)

Ejercicio 01.02:

HI = Σ_i=0..n 2ⁱ = 2ⁿ⁺¹-1

• CB: n = 0

Σ_i=0..0 2ⁱ = 1

2⁰⁺¹-1 = 2-1 = 1 OK

• PI: P(n)=>P(n+1)

Σ_i=0..n+1 2ⁱ = Σ_i=0..n 2ⁱ + 2ⁿ⁺¹ = (HI) 2ⁿ⁺¹-1 + 2ⁿ⁺¹ = 2 * 2ⁿ⁺¹-1 = 2ⁿ⁺²-1 OK