Please help me understand the steps

Question

Consider a population that grows according to the recursive ruleLn+1=Ln+26,with initial population L0=2.

What is the common difference for this arithmetic sequence? 

d=

Answer 1

L_n+1 = L_n + 26 (to get the next term of the sequence add 26 to the current term)

L₀ = 2 (given)

So,

L₁ = 2 + 26 = 28

L₂ = 28 + 26 = 54

L₃ = 54 + 26 = 80

...

Hopefully you notice that there is a common difference between each term of the sequence (26).

2, 28, 54, 80, ...

An explicit rule for the nth term (where n is 1,2,3,4...):

L_n = 2 + (n -1)26 = 2 + 26n - 26 = 26n - 24

Using the explicit rule:

L₁ = 26(1) - 24 = 2

L₂ = 26(2) - 24 = 52 - 24 = 28

L₃ = 26(3) - 24 = 78 - 24 = 54

L₄ = 26(4) - 24 = 104 - 24 = 80

The advantage to defining the arithmetic sequence explicitly is that you can determine any term simply by plugging in the number of the term and evaluating.