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The third term of an arithmetic sequence of positive numbers, in which the difference between the terms is not zero, is the geometric mean of the first and eleventh terms. What is the ratio of the second term to the first term of the sequence?

 

The geometric sequence is basically if you want to find the geo mean of a and b, that's sqrt(ab)
  

 Aug 6, 2020
 #1
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First term:  a

Second term:  a + d

Third term:  a + 2d

Eleventh term:  a + 10d

 

Third term is the geometric mean of the first term and the eleventh term:

     a + 2d  =  sqrt( a · (a + 10d) )     --->     (a + 2d)2  =  a(a + 10d)

                                                         a2 + 4ad + 4d2  =  a2 + 10ad

                                                                 4ad + 4d2  =  10ad

                                                                           4d2  =  6ad

                                                                             2d  =  3a

                                                                               d  =  (3/2)a

 

Ratio of the second term to the first term: 

     (a + d) / a     --->   ( a + (3/2)a ) / a  

                                       (2a + 3a) / (2a)            (mult num and den by 2)

                                                 5a / 2a

                                                    5/2

 Aug 6, 2020
 #2
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Thanks understood completely.

 Aug 6, 2020

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