is a recurring decimal because the number 2 is repeated infinitely. Question Find the sum of each of the geometric seriesįinding the sum of a Geometric Series to InfinityĬonverting a Recurring Decimal to a Fractionĭecimals that occurs in repetition infinitely or are repeated in period are called recurring decimals.įor example, 0.22222222. įinding the number of terms in a Geometric Progressionįind the number of terms in the geometric progression 6, 12, 24. Write down the 8th term in the Geometric Progression 1, 3, 9. Write down a specific term in a Geometric Progression To find the nth term of a geometric sequence we use the formula:įinding the sum of terms in a geometric progression is easily obtained by applying the formulas: The geometric sequence has its sequence formation: Note that after the first term, the next term is obtained by multiplying the preceding element by 3. The geometric sequence is sometimes called the geometric progression or GP, for short.įor example, the sequence 1, 3, 9, 27, 81 is a geometric sequence. The common ratio (r) is obtained by dividing any term by the preceding term, i.e., Dayįind a formula for the amount paid on day \(n\text\right)P\)) goes to pay down the principal.Geometric Progression, Series & Sums IntroductionĪ geometric sequence is a sequence such that any element after the first is obtained by multiplying the preceding element by a constant called the common ratio which is denoted by r. How much will you be paid for the job in total under Option 1?Ĭomplete Table7.19 to determine the pay you will receive under Option 2 for the first 10 days. You can be paid 1 cent the first day, 2 cents the second day, 4 cents the third day, 8 cents the fourth day, and so on, doubling the amount you are paid each day. You can be paid $500 per day or Option 2. Suppose you are hired for a one month job (30 days, working every day) and are given two options to be paid. There is an old question that is often used to introduce the power of geometric growth. Let \(Q(n)\) be the amount (in mg) of warfarin in the body before the \((n+1)\)st dose of the drug is administered.Įxplain why \(Q(2) = (5+Q(1)) \times 0.08\) mg. Assume that at the end of a 24 hour period, 8% of the drug remains in the body. The drug is absorbed by the body and some is excreted from the system between doses. Suppose warfarin is taken by a particular patient in a 5 mg dose each day. The level of warfarin has to reach a certain concentration in the blood in order to be effective. Warfarin is an anticoagulant that prevents blood clotting often it is prescribed to stroke victims in order to help ensure blood flow. In Example7.12, we see an example of a sequence that is connected to a sum. Many important sequences are generated by addition. Under what conditions does a geometric series converge? What is the sum of a convergent geometric series? What is a partial sum of a geometric series? What is a simplified form of the \(n\)th partial sum of a geometric series? Section 7.2 Geometric Series Motivating Questions Population Growth and the Logistic Equation.Qualitative Behavior of Solutions to DEs.An Introduction to Differential Equations.Physics Applications: Work, Force, and Pressure.Area and Arc Length in Polar Coordinates.Using Definite Integrals to Find Volume by Rotation and Arc Length.Using Definite Integrals to Find Area and Volume. Using Technology and Tables to Evaluate Integrals.The Second Fundamental Theorem of Calculus. Constructing Accurate Graphs of Antiderivatives.Determining Distance Traveled from Velocity.Using Derivatives to Describe Families of Functions.Using Derivatives to Identify Extreme Values.Derivatives of Functions Given Implicitly.Derivatives of Other Trigonometric Functions.Interpreting, Estimating, and Using the Derivative.The Derivative of a Function at a Point.
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