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Unformatted text preview: h ﬁrst term a and common ratio r = 0 is given by
an = arn−1 , n≥1 While the formal proofs of the formulas in Equation 9.1 require the techniques set forth in Section
9.3, we attempt to motivate them here. According to Deﬁnition 9.2, given an arithmetic sequence
with ﬁrst term a and common diﬀerence d, the way we get from one term to the next is by adding
d. Hence, the terms of the sequence are: a, a + d, a + 2d, a + 3d, . . . . We see that to reach the nth
term, we add d to a exactly (n − 1) times, which is what the formula says. The derivation of the
formula for geometric series follows similarly. Here, we start with a and go from one term to the
next by multiplying by r. We get a, ar, ar2 , ar3 and so forth. The nth term results from multiplying
a by r exactly (n − 1) times. We note here that the reason r = 0 is excluded from Equation 9.1 is
to avoid an instance of 00 which is an indeterminant form.4 With Equation 9.1 in place, we ﬁnally
have the tools required to ﬁnd an explicit formula for the nth term of the sequen...
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