Aim: What is the arithmetic and geometric sequence? Do Now: Find the first four terms in the sequence whose nth term is 4n + 3. 2. Find the first 5 numbers in a sequence whose nth term is 2n.

Each term after the first term is four more than the previous term 7, 11, 15, 19 Each term after the first term is four more than the previous term 4 4 4 The sequence an = 4n + 3 is an arithmetic sequence. An arithmetic sequence has consecutive terms with a common difference. In this case, the common difference is 4. For an arithmetic sequence, the recursive formula is: an = a1 + d(n – 1) Where d is the common difference

For the arithmetic sequence 100, 97, 94, 91,…, find : a) the common difference b) the 20th term of the sequence d = – 3 a20 = a1 + d(n – 1) = 100 + -3(20 – 1) = 100 + -3(19) = 100 – 57 = 43

The 4th term of an arithmetic sequence is 80 and the 12th term is 32. a. What is the common difference? b. What is the first term of the sequence? a. b. 80 = a1 + 3(-6), 80 = a1 – 18, a1 = 98

If 4 and 28 are the 1st and 5th terms of an arithmetic sequence, find the 2nd, 3rd and 4th terms of the sequence. Use the recursive formula to find d The numbers 10, 16 and 22 are called arithmetic means between 4 and 28. An arithmetic mean is the average of the term before it and the term after it.

a. Find the common difference b. Write the nth term formula 1. Given an arithmetic sequence 3,6,9,12,…, n = 8 a. Find the common difference b. Write the nth term formula c. Find the value of 8th term d = 3 an = 3 + 3(n – 1) a8 = 24 2. Write the first six terms of the arithmetic sequence that has 12 for the first term and 42 for the six term. 12, 18, 24, 30, 36, 42 3. Find four arithmetic means between 3 and 18 6, 9, 12, 15

This sequence is called a geometric sequence 2, 4, 8, 16, 32 Each term of the sequence is formed by multiplying the pervious term by 2, or we could say that the ratio of each term to the previous term is a constant,2. This sequence is called a geometric sequence A geometric sequence is a sequence such that for all n, there is a constant r such that The constant r is called common ratio The common ratio of the sequence 2,4,8,16,32 is 2.

The recursive formula of geometric sequence is an = a1rn-1 When the geometric sequence is written in terms of a1 and r, the terms of a geometric sequence are: The recursive formula of geometric sequence is an = a1rn-1 Is the sequence 4,12,36,108,324,… a geometric sequence? The ratio of any two consecutive terms is always 3, therefore the sequence is a geometric sequence with common ratio = 3

Write the first five terms of the geometric sequence whose first term is a1 = 3 and whose common ratio is r = 2. a1 = 3, a2 = 6, a3 = 12, a4 = 24, a5 = 48 Find the 12th term of the geometric sequence whose first three terms are 5, 15, 45. . .

Find the 15th term of the geometric sequence whose first term is 20 and whose common ratio is 1.05. 39.599 The 4th term of a geometric sequence is 125, and the 10th term is 125/64. Find the 14th term. 125/1024

We already knew the arithmetic mean between any two terms is the average of the two terms. The geometric mean between any two terms is the square root of the product of the two terms. The geometric mean between 4 and 100 is

Find four geometric means between 5 and 1215 This question can be rewritten as if the 1st and 6th terms of a geometric sequence are 5 and 1215. find the 2nd, 3rd, 4th and 5th terms. We want to find the missing terms in the sequence 5, a2, a3, a4, a5, 1215. use the formula to determine the common ratio r 15, 45, 135 and 405