Stitz-Zeager_College_Algebra_e-book

The resulting arc has a length of t units and

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Unformatted text preview: (d) −5 − 2x + x2 5 k=1 k=1 8 (−1) (−1)k ln(k ) (f) k k=1 k=3 4 k −1 (c) k+1 k 20 k −1 (b) (h) 25 (e) (3k + 5) (g) 20 (f) 0 5 2. (a) 17 2 (−1) k=1 6 x 2k − 1 (g) k=1 4 29 k−1 (d) (h) 2 k=1 k=1 3. (a) 305 7 9 13 (b) 99 4. (a) 5. $76,163.67 (b) (−1)k−1 k2 1 (x − 5)k 2k 1023 1024 (c) 3383 333 5809 (d) − 990 (c) 17771050 59049 9.3 Mathematical Induction 9.3 573 Mathematical Induction The Chinese philosopher Confucius is credited with the saying, “A journey of a thousand miles begins with a single step.” In many ways, this is the central theme of this section. Here we introduce a method of proof, Mathematical Induction, which allows us to prove many of the formulas we have merely motivated in Sections 9.1 and 9.2 by starting with just a single step. A good example is the formula for arithmetic sequences we touted in Equation 9.1. Arithmetic sequences are defined recursively, starting with a1 = a and then an+1 = an + d for n ≥ 1. This tells us that we start the sequence with a and we go from one term to the next by successively adding d. In symbo...
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