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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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