Question

Calcular:
S = 1 × 2 + 2 × 3 + 3 × 4 + ⋯ + 26 × 27

Answer 1

Respuesta: Sacar el valor de n

1 × 2 + 2 × 3 + 3 × 4 + ⋯ +n.(n+1)

Y tu quieres calcular

1 × 2 + 2 × 3 + 3 × 4 + ⋯ + 26 × 27

n=15

Fórmula:  

S = n . (n+1) . (n+2)  / 3

Reemplazas:

S=25x26x27

S=17550/ 3

S=5850

Answer 2

Answer:

Notice that

1 × 2 + 2 × 3 + 3 × 4 + ⋯ + 26 × 27

= Σk(k+1) , where k = 1 , 2,..., 26. But

Σk(k+1) = Σk^2 + Σk. Now, recall that

Σk^2 = n(n+1)(2n+1)/6 and Σk = n(n+1)/2.

Therefore, for k in {1,2,..,26}, you have

Σk(k+1) = 26(27)(53)/6 + 26(27)/2

= 6 201 + 351

= 6 552

That is

1 × 2 + 2 × 3 + 3 × 4 + ⋯ + 26 × 27

= 6 552.

and I'm done.