Exercise 5 Page 278

The common difference of the arithmetic sequence in this case is -2.

Recursive Formula: A(n)=A(n-1)-2, A(1)=9
Explicit Formula: A(n)=9-2(n-1)

Practice makes perfect

In order to write the formulas for the given arithmetic sequence, we first need to find the common difference. We can do this by subtracting the first term of the sequence from the second term. a_2-a_1=d ⇒ 7-9=- 2 The common difference is -2. Let's now find the recursive formula and the explicit formula one at a time.

Recursive Formula

To write the recursive formula, we will use the general form for a recursive formula of an arithmetic sequence. A(n)=A(n-1)+d In the above formula, n is the position of a term and d the common difference. By substituting the value of the common difference - 2 for d into this formula, we can create our recursive formula.

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

Substitute

d= - 2

A(n)=A(n-1)+( - 2)

AddNeg

a+(- b)=a-b

A(n)=A(n-1)-2

We can now write the recursive formula. Do not forget to write the first term! A(n)=A(n-1)-2, A(1)=9

Explicit Formula

To write the explicit formula, we will use the general form of an explicit formula. A(n)=A(1)+(n-1)d In the above formula, n and d represent the same as in the recursive formula. By substituting A(1)=9 and d=-2, we can create our explicit formula.

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

SubstituteII

A(1)= 9, d= - 2

A(n)= 9+(n-1)( - 2)

CommutativePropMult

Commutative Property of Multiplication

A(n)=9+(- 2)(n-1)

AddNeg

a+(- b)=a-b

A(n)=9-2(n-1)

Therefore, the explicit formula is A(n)=9-2(n-1).