Question

Given the formula for an arithmetic sequence f(7) = f(6) + 4 written using a recursive formula, write the sequence using an arithmetic formula

Of(7) = f(1) + 24

1(7) = f(1) + 20

Of(7) = f(1) + 12

Of(7) = f(1) + 4

Answer 1

f(7) = f(1) + 24

Explanation:

Arithmetic explicit formula has the form:

f(n) = f(1) + (n-1)*d

Arithmetic recursive formula has the form:

f(n) = f(n-1) + d

Given the expression: f(7) = f(6) + 4, it can be seen it's expressed in the recursive form with n = 7, n-1 = 6 and d = 4. Using these values in the explicit formula we get:

f(7) = f(1) + 6*4

f(7) = f(1) + 24

Answer 2

Of(7) = f(1) + 24

Explanation:

Since this Arithmetic Sequence can be written recursively as a function, then we can write the whole sequence, by adding the common difference to the previous function. So writing it as an Arithmetic formula is (placing an example, with a common difference of 4 units):

`\left.\begin{matrix}n &1&2&3&4&5&6 &7\\f(n)&2&6&10 & 14&18&22&26 \end{matrix}\right|\\f(n)=f(n-1)+d\:\: (Recursive \: Formula)\Rightarrow a_(n)=a_(n-1)+4\: \: \: (Arithmetic\: Formula)\\Of(7)=f(1)+6*4\Rightarrow a_(7)=a_(1)+(7-1)4\Rightarrow a_(7)=a_(1)+24\\`