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

Answer:

Of(7) = f(1) + 24

Step-by-step 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):

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

Answer 2

Answer:

f(7) = f(1) + 24

Step-by-step 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