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**Unformatted text preview: **,k≥0
2k + 1 4. 1 3 9 27
,− , ,− ,...
2 4 8 16 2
Math fans will delight to know we are basically talking about the ‘countably inﬁnite’ subsets of the real number
line when we do this.
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Sequences which are both arithmetic and geometric are discussed in the Exercises. 9.1 Sequences 555 Solution. A good rule of thumb to keep in mind when working with sequences is “When in doubt,
write it out!” Writing out the ﬁrst several terms can help you identify the pattern of the sequence
should one exist.
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1. From Example 9.1.1, we know that the ﬁrst four terms of this sequence are 1 , 5 , 27 and 125 .
39
81
To see if this is an arithmetic sequence, we look at the successive diﬀerences of terms. We
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ﬁnd that a2 − a1 = 9 − 1 = 2 and a3 − a2 = 27 − 9 = 10 . Since we get diﬀerent numbers,
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9
27
there is no ‘common diﬀerence’ and we have established that the sequence is not arithmetic.
To investigate whether or not it is geometric, we compute the ratios of successive terms. The
ﬁrst three r...

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