The ratio of 6th and 8th term of an A.P. is 7:9. Find the ratio of 9th term to 13th term.

The ratio of 6th and 8th term of an A.P. is 7:9. Find the ratio of 9th term to 13th term.
Solution :
t6 : t8  = 7 : 9
t6 / t8  = 7 / 9
(a + 5d) / (a + 7d)  = 7/9
9(a + 5d)  = 7(a + 7d) 
9a + 45d  = 7a + 49d
9a - 7a + 45d  - 49d = 0
2a - 4d  = 0
a - 2d  = 0
a  = 2d
ratio of 9th term to 13th term
t9 : t13  = (a + 8d) / (a + 12d)
  =  (2d + 8d) / (2d + 12d)
  =  10d / 14d
  =  5 / 7
  =  5 : 7
Hence the ratio of 9th term to 13th term is 5 : 7.

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