SOLUTION: if the first, fifth and tenth terms of an arithmetic sequence are in geometric progression and the sun of the second and eight terms is 20, find the first term and the non zero co

Algebra ->  Sequences-and-series -> SOLUTION: if the first, fifth and tenth terms of an arithmetic sequence are in geometric progression and the sun of the second and eight terms is 20, find the first term and the non zero co      Log On


   

Question 732668: if the first, fifth and tenth terms of an arithmetic sequence are in geometric progression and the sun of the second and eight terms is 20, find the first term and the non zero common difference
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
a= first term
d= common difference (d%3C%3E0)

a%2Bd= second term
a%2B%288-1%29d=a%2B7d= 8th term
second term + 8th term =a%2Bd%2Ba%2B7d=2a%2B8d=2%28a%2B4d%29=20
2%28a%2B4d%29=20 --> a%2B4d=10 <--> a=10-4d

fifth term =a%2B%285-1%29d=a%2B4d=10
tenth term =a%2B%2810-1%29d=a%2B9d
The first, fifth and tenth terms are in geometric progression, so the common ratio of that geometric progression is
10%2Fa=%28a%2B9d%29%2F10 --> 10%2A10=a%28a%2B9d%29 --> 100=a%28a%2B9d%29
Substituting a=10-4d we get
100=%2810-4d%29%2810-4d%2B9d%29 --> 100=%2810-4d%29%2810%2B5d%29 --> 100=100%2B10d-20d%5E2 --> 0=10d-20d%5E2 --> 0=10d%281-2d%29
Since d%3C%3E0 it must be 1-2d=0 --> {{2d=1}}} --> highlight%28d=1%2F2%29
and a=10-4%281%2F2%29%7D%7D+--%3E+%7B%7B%7Ba=10-2 --> highlight%28a=8%29