You have the following data set:
n | f(n)
1 5
2 10
3 20
4 40
In order to determine the resurve function associated to f(n), you consider that each term of f(n) is twice the previous one. Thus you have:
f(n) = 2*f(n-1) "this is the recursive function"
The explicit function is obtained by coinsider the values of n letf side of the table. You can notice that f(n) is multiple of products of 2's, that is:
f(n) = 5(2)ⁿ⁻¹ "this is the explicit function"
In fact, you obtain for the values of n:
f(1) = 5(2)¹⁻¹ = 5(2)⁰ = 5(1) = 5
f(2) = 5(2)²⁻¹ = 5(2)¹ = 5(2) = 10
f(3) = 5(2)³⁻¹ = 5(2)² = 5(4) = 20
f(4) = 5(2)⁴⁻¹ = 5(2)³ = 5(8) = 40
which coincides perfectly with the results of the table