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# Plz Help

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(a) Determine all nonnegative integers r such that it is possible for an infinite arithmetic sequence to contain exactly r terms that are integers. Prove your answer.

(b) Determine all nonnegative integers r such that it is possible for an infinite geometric sequence to contain exactly r terms that are integers. Prove your answer.

Sep 13, 2019

#1
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$$a)~a_n = a_0 + n d,~n \in \mathbb{Z^+},~d,a_0 \in \mathbb{R}\\ \text{Suppose there are 2 or more integers in the sequence, at say k_1,~k_2}\\ a_0 + k_1 d \in \mathbb{Z} \text{ and }a_0 + k_2 d \in \mathbb{Z}\\ (k_1-k_2)d \in \mathbb{Z}\\ m(k_1 - k_2)d \in \mathbb{Z},~m \in \mathbb{Z}\\ a_0 + k_1 d + m(k_1-k_2)d \in \mathbb{Z}\\ \text{This corresponds to index }\\ (m+1)k_1 - mk_2\\ \text{and thus there are infinitely many integers in the sequence as m \in \mathbb{Z}}$$

$$\text{On the other hand we can easily form a sequence with 1 integer by having \\a_0 \in \mathbb{Z},~d \not \in \mathbb{Q}}$$

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Sep 14, 2019
#2
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$$b)~a_n = a_0 d^n,~n \in \{0\}\cup \mathbb{N},~a_0, d\in \mathbb{R}\\ \text{If we let a_0 = \left(\dfrac 1 d\right)^{r-1},~\dfrac 1 d \in \mathbb{N}}\\ \text{we end up with exactly r integers in the sequence}$$

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Sep 14, 2019