SOLUTION: The ninth term of an arithmetic progression is 52 and the sum of the first twelve terms is 414.Find the first term and the common difference.

Algebra ->  Sequences-and-series -> SOLUTION: The ninth term of an arithmetic progression is 52 and the sum of the first twelve terms is 414.Find the first term and the common difference.      Log On


   

Question 936199: The ninth term of an arithmetic progression is 52 and the sum of the first twelve terms is 414.Find the first term and the common difference.
Answer by srinivas.g(540) About Me  (Show Source):
You can put this solution on YOUR website!
formula:
nth term =a+(n-1)d
sum of n terms = %28n%2F2%29%282a%2B%28n-1%29d%29
where d = common difference , a = first term
9 th term = a+(9-1)d
52 = a+8d...................eq(1)
sum of fisrt 12 terms = 414
%2812%2F2%29%282a%2B%2812-1%29d%29+=414
6%282a%2B11d%29=414
divide with 6 on both sides
+6%282a%2B11d%29%2F6+=414%2F6
2a%2B11d+=+69.............eq(2)
we need to solve eq(1) & eq(2)
Solved by pluggable solver: SOLVE linear system by SUBSTITUTION
Solve:
+system%28+%0D%0A++++1%5Ca+%2B+8%5Cd+=+52%2C%0D%0A++++2%5Ca+%2B+11%5Cd+=+69+%29%0D%0A++We'll use substitution. After moving 8*d to the right, we get:
1%2Aa+=+52+-+8%2Ad, or a+=+52%2F1+-+8%2Ad%2F1. Substitute that
into another equation:
2%2A%2852%2F1+-+8%2Ad%2F1%29+%2B+11%5Cd+=+69 and simplify: So, we know that d=7. Since a+=+52%2F1+-+8%2Ad%2F1, a=-4.

Answer: system%28+a=-4%2C+d=7+%29.