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The first term of an arithmetic sequence is 1, another term of the sequence is 91 and all of the terms of the sequence are integers. How many distinct arithmetic sequences meet these three conditions?

Sep 22, 2019

#1
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The sequences with the following terms are:
First term = 1
nth term    = 91
All term must be integers.
Common difference          91 as a term
1                                          91st. term
2                                          46th term
3                                          31st term
5                                           19th term
6                                           16th term
9                                            11th term
10                                          10th term
15                                           7th term
18                                            6th term
30                                            4th term
45                                            3rd term
90                                            2nd term
As you can see, there are 12 sequences which start at 1 and have 91 as one of terms and are all integers.

Sep 22, 2019
#2
0

Thank you so much

Sep 22, 2019