82, 77, 72, 67, 62 The recursive pattern in the sequence above is latex-_ (be sure to show if it is positive or negative). The 4th term in the sequence is . An explicit formula that I can use to write for this sequence is latex-_ . The value of the 100th term is .

a. It is a negative pattern

b. Fourth term is 67

c. Explicit formula is Tn = 87 - 5n where n is the number of the term.

d. The value of the 100th term is -413

Step-by-step explanation:

1.Firstly, we want to know if the pattern is positive or negative.

Looking at the pattern, we can see our first term as 82, second as 77, third as 72.

What this show is that the pattern is decreasing as we move on from one term to another. So we can say it is a negative recursive pattern.

2. We are asked which term is the 4th.

This is quite obvious from the spelt out pattern.

82 77 72 67

1234

67 is the fourth term.

3. We want an explicit formula for the sequence.

What this question is asking us is a formula such that we can use at any point in time to calculate the value of the term at that point.

Firstly we have the first term as 82.

We need the common difference.

Mathematically this is the second term minus the first term which is same as the third term minus the second term.

That would be; 77-82 = 72-77 = -5

Mathematically the nth term of an arithmetic sequence is;

Tn = a + (n-1) d

Where in this case a is 82 and d is -5

Substituting these values, we have;

Tn = 82 + (n-1)-5

Tn = 82 -5n + 5

Tn = 82 + 5 -5n

Tn = 87 -5n

Where Tn is the term we are looking for while n is the number of the term.

4. Value of the 100th term

Simply substitute the value of 100 for n in the explicit formula.

That would be;

87 -5(100) = 87-500 = -413

a = 3.14

b = 18

c = 18/5 or 3.6 or 3 3/5

step-by-step explanation: