UNIT 1 - ARITHMETIC & GEOMETRIC SEQUENCES Task #10 – Derive Formula for Infinite Geometric Series Common Core: HS.A-SSE.B.4, HS.F-BF.A.2 MA40: ALGEBRA 2 Name: Period: INVESTIGATION Convergence – An infinite series is convergent if the sequence of its partial sums S1, S2 , S3 , ... approaches a given number. Divergence – An infinite series is divergent if the sequence of its partial sums S1, S2 , S3 , ... do not approach a given number. I) Investigate the series represented by the summation notation by completing the table. Use your calculator to complete the table. n 1 i1 3 i 1 2 This is a(n) _______________ series. We will use the formula ____________________ to find the sum of the first n terms of the series. S n is the ________________________________________. n is the __________________________. a1 is the _______________ & equals ___________. r is the ________________ & equals _________. n Use x on calculator. Input value. rn Use Y1 on calculator 2nd Y Sn n Expand the series Y2 on calculator 2nd Y screen. Use 1 2 3 i 1 . i 1 Consider the ending terms of the series. screen. n 1 n2 n3 1 2 1 3 S2 2 2 1 3 9 S3 2 2 2 S1 1 3 9 ... 2 2 2 1 3 9 S9 ... n9 2 2 2 1 3 9 S10 ... n 10 2 2 2 Question #1 – Consider column 4. What do you notice about the last term of each series as n increases? Do you think this trend will continue? n 8 S8 Question #2 – Consider column 3. As the number of terms n increases, what do you notice about the sum S n . Task #10 – Derive Formula for Infinite Geometric Series – (continued) Question #3 – Will the values in column 3 continue to grow without bound or will they eventually “equal” some value? Does the series represented by the summation notation represent a convergent or divergent series? Explain. II) Investigate the series represented by the summation notation by completing the table. Use your calculator to complete the table. 1 3 2 i 1 n i 1 This is a(n) ______________ series. We will use the formula ___________________ to find the sum of the first n terms of the series. S n is the ________________________________________. n is the __________________________. a1 is the _______________ & equals ___________. r is the ________________ & equals _________. n Use x on calculator. Input value. rn Use Y1 on calculator 2nd Y Sn 1 3 2 i 1 n Expand the series Y2 on calculator 2nd Y screen. Use i 1 . Consider the ending terms of the series. screen. n 1 S3 3 n2 S2 3 n3 3 2 3 3 S3 3 2 4 3 3 ... 2 4 3 3 S12 3 ... n 12 2 4 3 3 S13 3 ... n 13 2 4 Question #4 – Consider column 4. What do you notice about the last term of each series as n increases? Do you think this trend will continue? n 11 S11 3 Question #5 – Consider column 3. As the number of terms n increases, what do you notice about the sum S n . Question #6 – Will the values in column 3 continue to grow without bound or will they eventually “equal” some value? Does the series represented by the summation notation represent a convergent or divergent series? Explain. Task #10 – Derive Formula for Infinite Geometric Series – (continued) Question #7 – Consider column 2. What do you notice about the value of r n as the number of terms n increases? Do you think this has anything to do with your answer to Question #6? Explain. DEVELOPING THE MATH CONCEPTS & TERMS Deriving the Formula for the Sum of an Infinite Geometric Series. Finding the Sums of Infinite Geometric Series 1) Find 3 0.7 i 1 i 1 1 1 1 2) Find the sum of 1 ... 4 16 64 Finding the Common Ratio 3) An infinite geometric series with first term a1 4 has the sum of 10. What is the common ratio of the series? Task #10 – Derive Formula for Infinite Geometric Series – (continued) Using an infinite Series as a Model 3) A ball is dropped from a height of 10 feet. Each time it hits the ground, it bounces to 80% of its previous height. Find the total distance traveled by the ball. 4) Writing a Repeating Decimal as a Fraction Write 0.181818... as a fraction Summary: