Geometric Sequences and Series: Finding Sums of Finite Geometric Sequences, Lecture notes of Geometry

Download Geometric Sequences and Series: Finding Sums of Finite Geometric Sequences and more Lecture notes Geometry in PDF only on Docsity! Accel Math III Unit #3: Sequences & Series Lesson #3: Finite Geometric Sequences and Series MA3A9:Students will use sequences and series. b. Recognize and use simple arithmetic and geometric sequences. e. Find and apply the sums of finite and, where appropriate, infinite arithmetic and geometric series. f. Use summation notation to explore series. g. Determine geometric series and their limits. EQ: What is the formula to find the sum of a finite geometric sequence? Recall: • Geometric Sequence --- ordered list of numbers with a common __________called _______ Ex 1. Determine if each sequence is geometric. Justify your answer. a. 2, 8, 12, 48, 52, … b. -8, 4, -2, 1, … Write the explicit formula for finding the nth term of a geometric sequence. an = _________________ Ex 2. Write the explicit formula of the geometric sequence whose initial term is 3 and whose common ratio is 2. Then find the first five terms. an = _________________ a1 = ___ a2 = ___ a3 = ___ a4 = ___ a5 = ___ Ex 3. Write the explicit formula of the geometric sequence whose first term is a1 = 20 and has a common ratio of r = 1.05. Then find the 15th term of the sequence. an = _________________ a15 = ___ How to Find the Sum of a Finite Geometric Sequence: GOAL: We are looking for Sn 1. Begin with the Sum of an Infinite Geometric Sequence: Sn = a1 + a1r + a1r 2 + a1r 3 + … + a1r n-1 2. Multiplying both sides by r: WHY CAN YOU DO THIS? rSn = a1r + a1r 2 + a1r 3 + a1r 4 … + a1r n-1+ a1r n RECALL: SOLVING SYSTEMS OF EQUATIONS 3. Subtracting these two equations: Sn = a1 + a1r + a1r 2 + a1r 3 + … + a1r n-1 - rSn = a1r + a1r 2 + a1r 3 + …+ a1r n-1+ a1r n Sn - rSn = a1 - a1r n RECALL: What is the GOAL? 4. Now Factor out Sn : Sn(1 – r) = a1(1 – rn) 5. Finally, solve for Sn ( )1 1 1 n n a r S r − = − r ≠ 1 WHY? Ex 6. Use the summation formula to find the sum ( )∑ = 12 1 3.04 n n a. n = ____ a1 = ______ r = ______ rn = ________ ( )1 1 1 n n a r S r − = − = _________________ b. n = ____ a1 = ______ r = ______ rn = ________ ( )1 1 1 n n a r S r − = − = _________________ Assignment: p. 644 ODDS #1 – 41, 55 - 67