Unit #4: Sequences & Series Accel Precalc Unit #4: Sequences & Series Lesson #4: Infinite Geometric Sequences and Series EQ: What is the formula to find the sum of an infinite geometric sequence?

∞ 1 When the ratio is greater than 1, the terms in the sequence will get _______ and _______. larger larger If you add larger and larger numbers forever, you will get___ for an answer. ∞ So, we don't deal with infinite geometric series when the ratio is greater than ___. 1

1 The ratio can't equal ___ because then the series wouldn't be geometric (WHY?) and the sum formula would have division by ___. The only case left, then, is when the ratio is ______ than 1. less

List the first 10 terms for this sequence. ¼ 1/8 1 ½ 1/16 1/32 1/64 1/128 1/256 1/512 As the sequence goes on, the terms are getting ________ and ________, approaching ____. smaller smaller

 So, if you replace rn with ___ in the summation formula, the 1 - rn part just becomes ____, and the numerator just becomes ____. 1 a1

Day 34 Agenda: DG 15 --- 10 minutes

***If |r| > 1, then finding the sum is NOT POSSIBLE!

2.4 0.6

0.1 0.01 0.001 .1 1 .1 2 .1 3 3(0.1)n-1 0.1 3

= 0.5 + 0.05 + 0.005 + 0.0005 + … = 5(0.1) + 5(0.01) + 5(0.001) + 5(0.0001) + … = 5(0.1)1 + 5(0.1)2 + 5(0.1)3 + 5(0.1)4 + … OR

= 0.47 + 0.0047 + 0.000047 + … = 47(0.01) + 47(0.0001) + 47(0.000001) + … = 47(0.01)1 + 47(0.01)2 + 47(0.01)3 + … OR

= 0.1 + 0.06 + 0.006 + 0.0006 + … = 0.1 + 0.6(0.1) + 0.6(0.01) + 0.6(0.001) +… = 0.1 + 0.6(0.1) + 0.6(0.1)2 + 0.6(0.1)3 +… OR CHECK:

At --- P0 --- r --- n --- t --- Recall: final amount of money after t   P0 --- r --- n --- t --- final amount of money after t initial amount of money; principle annual interest rate (decimal) times per year interest is calculated total number of times interest is calculated

Ex. A deposit is made of $50 on the first day of each month in a savings account that earns 6% compounded monthly. What is the balance of this annuity at the end of 2 years?

Total Balance will be sum of the balances after the 24 deposits. Use the sum formula for a finite geometric series. 24 50(1.005)1 This is the LAST deposit made. It will earn interest for 1 month.

This account will have a balance of $1277.95 at the end of 2 years. Ex. A deposit is made of $50 on the first day of each month in a savings account that earns 6% compounded monthly. What is the balance of this annuity at the end of 2 years? This account will have a balance of $1277.95 at the end of 2 years.

Assignment: p. 645 ODDS #69 – 91