Given the sequence 1 3 7 13 21 provide a 1 recursive definit

Given the sequence 1, 3, 7, 13, 21... provide a:

1) recursive definition and

2) explicit definition (in terms of n)

Solution

Explicit definition = It gives the value of a specified term based on its position.

Recursive definition = It gives the value of a specified term based on the previous term.

now,

(1)

T₁ = 1

T₂ = 3

T₃ = 7

T₄ = 13

T₅ = 21

T₆ = 31

T₂ - T₁ = 2 => U₁

T₃ - T₂ = 4 => U₂

T₄ - T₃ = 6 => U₃

T₅ - T₄ = 8 => U₄

T₆ - T₅ = 10 => U₅

U₂ - U₂ = 2

U₃ - U₂ = 2

U₄ - U₃ = 2

U₅ - U₄ = 2

So,

U_n -U_n-1 = 2

U_n-1 = T_n - T_n-1

U_n = T_n+1 - T_n

U_n - U_n-1 = T_n+1 - T_n - T_n + T_n-1 = 2

T_n+1 - 2T_n + T_n-1 = 2

T_n+1 = 2 + 2T_n - T_n-1

(2)

One way of looking at the sequence is to say that the difference between successive numbers increases by two.

The general term in the sequence is given by

t_n = n² - n + 1 for n = 1,2,3,...

or, equivalently, t_n = (n³ + 1)/(n+1)