Exercise 68 Page 306

Practice makes perfect

a With an explicit equation you can find any term by knowing its position in the sequence. On the other hand, a recursive equation requires you to know the previous term to the one you want to determine. Therefore, unless you want to calculate each of the 49 terms that come before the last known term, choose an explicit equation.

b To determine if the sequence is arithmetic or geometric, we have to find out if there is a common difference or a common ratio between consecutive terms.

We have a common difference between consecutive terms, which means this is an arithmetic sequence. The formula for an arithmetic sequence follows a certain format. a_n= mn+a_0 In the formula n is the term number, m is the common difference, and a_0 is the zeroth term. We have already determined that the common difference is m= 4. a_n= 4n+a_0 To find the zeroth term we can, for example, substitute a_1=- 3 in the formula and solve for a_0.

a_n=4n+a_0

Substitute

n= 1

a_1=4( 1)+a_0

MultByOne

a * 1=a

a_1=4+a_0

Substitute

a_1= - 3

- 3=4+a_0

SubEqn

LHS-4=RHS-4

- 7=a_0

RearrangeEqn

Rearrange equation

a_0= - 7

Now we can complete the equation. a_n=4n+(- 7) ⇔ a_n=4n-7

c To calculate the 50^(th) term, we have to substitute n=50 in the equation we found in Part B and simplify.

a_n=4n-7

Substitute

n= 50

a_(50)=4( 50)-7

Multiply

a_(50)=200-7

SubTerm

Subtract term

a_(50)=193

The 50^(th) term of the sequence is 193.

d Like in Part B, we have to determine if the sequence is arithmetic or geometric.

We have a common difference between consecutive terms, which means this is an arithmetic sequence. Let's substitute this into the general formula. We have already determined that the common difference is m= - 13. a_n= - 1/3n+a_0 To find the zeroth term we can, for example, substitute a_1=3 in the formula and solve for a_0.

a_n=- 1/3n+a_0

Substitute

n= 1

a_1= - 1/3( 1)+a_0

MultByOne

a * 1=a

a_1=- 1/3+a_0

Substitute

a_1= 3

3=- 1/3+a_0

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Solve for a_0