862 CHAPTER 12 Sequences; Induction; the Binomial Theorem Figure 12 Figure 10 Figure 11 Algebraic Solution (a) ∑ ∑ ( ) ( ) = = ⋅ + = ⋅ = = = k k 3 3 3 5 5 1 2 3 15 45 k k 1 5 1 5 Property (1) Formula (6) (b) ∑ ∑ ∑ ( ) ( ) ( ) + = + = + + ⋅ = + = = = = k k 1 1 10 10 1 2 1 10 3025 10 3035 k k k 1 10 3 1 10 3 1 10 2 Property (2) Formulas (8) and (5) (c) ∑ ∑ ∑ ∑ ∑ ∑ ∑ ( ) ( ) ( )( ) ( ) − + = − + = − + = + ⋅ + − ⋅ + + ⋅ = − + = = = = = = = = k k k k k k 7 2 7 2 7 2 24 24 1 2 24 1 6 7 24 24 1 2 2 24 4900 2100 48 2848 k k k k k k k 1 24 2 1 24 2 1 24 1 24 1 24 2 1 24 1 24 (d) Notice that the index of summation starts at 6. Use property (4) as follows: k k k k 4 4 4 4 20 21 41 6 5 6 11 6 4 2870 55 11,260 k k k k 6 20 2 6 20 2 1 20 2 1 5 2 ∑ ∑ ∑ ∑ ( ) [ ] = = − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = ⋅ ⋅ − ⋅ ⋅ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = − = = = = = Graphing Solution (a) Figure 10 shows the solution using a TI-84 Plus CE graphing calculator. So ∑( ) = = k3 45. k 1 5 (b) Figure 11 shows the solution using GeoGebra. So ∑( ) + = = k 1 3035. k 1 10 3 (d) Figure 13 shows the solution using a TI-84 Plus CE graphing calculator. (c) Figure 12 shows the solution using Desmos. NOTE: To get sigma, type “sum” in the command bar. Now Work PROBLEM 75 5 Solve Annuity and Amortization Problems Using Recursive Formulas In Section 5.7 we developed the compound interest formula, which gives the future value when a fixed amount of money is deposited in an account that pays interest compounded periodically. Often, though, money is invested in small amounts at periodic intervals. An annuity is a sequence of equal periodic deposits. The periodic deposits may be made annually, quarterly, monthly, or daily. Properties (2) and (3) Property (1) Formulas (7), (6), (5) ↑ ↑ ↑ Property (1) Property (4) Formula (7) Figure 13
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