A sequence is defined by an = n3 – 6n2 + 11n – 6, n ∈ N. Show that the first three terms of the sequence are zero and all other terms are positive.
Given,
an = n3 – 6n2 + 11n – 6, n ∈ N
We can find first three terms of sequence by putting the values of n form 1 to 3.
When n = 1:
a1 = (1)3 – 6(1)2 + 11(1) – 6
⇒ a1 = 1 – 6 + 11 – 6
⇒ a1 = 12 – 12
⇒ a1 = 0
When n = 2:
a2 = (2)3 – 6(2)2 + 11(2) – 6
⇒ a2 = 8 – 6(4) + 22 – 6
⇒ a2 = 8 – 24 + 22 – 6
⇒ a2 = 30 – 30
⇒ a2 = 0
When n = 3:
a3 = (3)3 – 6(3)2 + 11(3) – 6
⇒ a3 = 27 – 6(9) + 33 – 6
⇒ a3 = 27 – 54 + 33 – 6
⇒ a3 = 60 – 60
⇒ a3 = 0
This shows that the first three terms of the sequence is zero.
When n = n:
an = n3 – 6n2 + 11n – 6
⇒ an = n3 – 6n2 + 11n – 6 – n + n – 2 + 2
⇒ an = n3 – 6n2 + 12n – 8 – n + 2
⇒ an = (n)3 – 3×2n(n – 2) – (2)3 – n + 2
{(a – b)3 = (a)3 – (b)3 – 3ab(a – b)}
⇒ an = (n – 2)3 – (n – 2)
Here, n – 2 will always be positive for n > 3
∴ an is always positive for n > 3