Definition of Geometric Sequence Geometric Sequences and Series Group 11 B Algebra II Academia Santa Rosa Prof. Eddie Ortiz Roman, Ph.D Principal Dr. Lorrie Cuevas Definition of Geometric Sequence

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

The nth Term of a Geometric Sequence The nth term of a geometric sequence has the form an = a1rn - 1 where r is the common ratio of consecutive terms of the sequence. a1 = 15 15, 75, 375, 1875, . . . a2 = 15(5) a3 = 15(52) a4 = 15(53) The nth term is 15(5n-1). The nth Term of a Geometric Sequence

Example 3. Find the 15th 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

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

The Sum of a Finite Geometric Sequence The sum of a finite geometric sequence is given by 5 + 10 + 20 + 40 + 80 + 160 + 320 + 640 = ? n = 8 a1 = 5 The Sum of a Finite Geometric Sequence

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. If |r| < 1, then the infinite geometric series a1 + a1r + a1r2 + a1r3 + . . . + a1rn-1 + . . . has the sum Definition of Geometric Series

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 3 + 0.3 + 0.03 + 0.003 + …,

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.