Exercise 3 Page 192

Practice makes perfect

The statement says that f(1) = -8; f(n) = -4(n-1) for all n ≥ 2 is a valid recursive rule for the sequence -8, -4, 0, 4, 8, 12, ... Let's start by recalling the general form of a recursive rule.

f(n) = f(n-1)+ d for all n ≥ 2 In this equation, the first term of the sequence f(1) is supposed to be known, and d is the common difference. Notice that the rule given does not have the required form. Therefore, this is not even a recursive rule! This statement is false.

To know if f(n) = -8+4(n-1) is a valid explicit rule for the sequence -8, -4, 0, 4, 8, 12, ... we can compare it to the general from of an explicit rule.

f(n) = f(1) + d(n-1)In this equation f(1) is the first term of the sequence, and d is the common difference. We can compare the general explicit rule to the given one to identify its parameters. f(n) = f& (1) + d (n-1) & ↓ ↓ f(n) = & -8 + 4 (n-1) Notice that the given rule corresponds to a sequence starting at -8 just as the sequence we want to describe. We just need to confirm if the common differences match. Let's find the difference between consecutive terms to confirm this. -8 +4 ⟶ -4 +4 ⟶ 0 +4 ⟶ 4 +4 ⟶8 +4 ⟶12 We can see that the common difference is 4. Therefore, the explicit rule f(n) = -8+4(n-1) corresponds to the sequence -8, -4, 0, 4, 8, 12, ... and hence, this statement is true.

As we saw in Part B, the sequence can be described by the explicit rule f(n) = -8+4(n-1). We can substitute n=10 to verify if the 10^(th) term is 28.

f(n) = -8+4(n-1)

Substitute

n= 10

f( 10) = -8+4( 10-1)

SubTerm

Subtract term

f(10) = -8+4(9)

Multiply

f(10) = -8+36

AddTerms

Add terms