Geometric Sequences and Series A sequence is geometric if the ratios of consecutive terms are the same. 2, 8, 32, 128, 512,... Definition of Geometric Sequence geometric sequence The common ratio, r, is 4.

The nth term of a geometric sequence has the form a n = a 1 r n - 1 where r is the common ratio of consecutive terms of the sequence. The nth Term of a Geometric Sequence 15, 75, 375, 1875,... a 1 = 15 The nth term is 15(5 n-1 ). a 2 = 15(5) a 3 = 15(5 2 ) a 4 = 15(5 3 )

Example 3. Find the 15 th term of the geometric sequence whose first term is 20 and whose common ration is 1.05.

Example 4. Find a formula for the nth term of the following geometric sequence. What is the ninth term of the sequence? 5, 15, 45, … Find the common ratio

The sum of the first n terms of a sequence is represented by summation notation. Definition of Summation Notation index of summation upper limit of summation lower limit of summation

The Sum of a Finite Geometric Sequence The sum of a finite geometric sequence is given by = ? n = 8 a 1 = 5

Write out a few terms. If the index began at i = 0, you would have to adjust your formula

Definition of Geometric Series The sum of the terms of an infinite geometric sequence is called a geometric series. a 1 + a 1 r + a 1 r 2 + a 1 r a 1 r n If |r| < 1, then the infinite geometric series has the sum

Example 7. Use a graphing calculator to find the first six partial sums of the series. Then find the sum of the series. Use the formula for the sum of an infinite series to find the sum.

Example 8. Find the sum of …,

Example 9. A deposit of $50 is made on the first day of each month in a savings account that pays 6% compounded monthly. What is the balance at the end of 2 years? This type of savings plan is called an increasing annuity. The first deposit will gain interest for 24 months, and its balance will be The second deposit will gain interest for 23 months The last deposit will gain interest for only one month The total balance will be the sum of the balances of the 24 deposits.

Homework Page even, odd, 33, 35, 40, 42, even