find the first term and the common diffrence

Question

the difference between the tenth term and the seventh term of an arithmatic sequence is -30. if the twelf term divided by the sixth teem is 3, find the first term and the common difference.

Answer 1

If a is the first term of an AP and p the common difference, then the 10th term is a+9p and the 7th is a+6p. If the difference is 30 then 3p=30 or -30 so p=10 or -10. The 12th term is a+11p and the 6th is a+5p, so (a+11p)/(a+5p)=3. Therefore, a+11p=3(a+5p)=3a+15p, so 2a+4p=0 or a+2p=0. p is 10 or -10, so a=-20 or 20.