Is the Sequence Arithmetic? Yes. Explanation: See solution. Recursive Formula: A(n)=A(n-1)-4, A(1)=- 3 Explicit Formula: A(n)=-3+(n-1)(- 4)
Practice makes perfect
When a sequence is arithmetic, the difference between two consecutive terms is constant. Examining our sequence we can see that this is the case.
-3 +( - 4) ⟶ -7 +( - 4) ⟶ -11 +( - 4) ⟶ -15 +( - 4) ⟶ ...
Since the difference between consecutive terms is constant and equal to - 4, we have an arithmetic sequence with common difference - 4. We will use this information to write both the recursive and the explicit formula.
Recursive Formula
Let's recall the general form of the recursive formula for an arithmetic sequence.
A(n)=A(n-1)+d
In the above formula, n is the term number and d the common difference. By substituting d= - 4 into this formula, we can create our recursive rule.
To complete the recursive formula, we need to state the value of the first term, which in this case is A(1)= - 3.
Recursive Formula
A(n)=A(n-1)-4, A(1)= - 3
Explicit Formula
Now, let's recall the general form of an explicit formula.
A(n)=A(1)+(n-1)d
In the above formula, n, d, and A(1) are the term number, the common difference, and the first term, respectively. By substituting A(1)= - 3 and d= - 4 into this rule, we can create our explicit formula.
