SOLUTION: Hello, How can i find the first and sum of the first 14 terms of an arithmetic sequence if i only know that the sum of the 3rd and 7th term is 24 and difference between the 6th

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Question 936170: Hello,
How can i find the first and sum of the first 14 terms of an arithmetic sequence if i only know that the sum of the 3rd and 7th term is 24 and difference between the 6th and 4th is 4?
Thanks alot :)
Answer by srinivas.g(540) About Me  (Show Source):
You can put this solution on YOUR website!
formula for n th term ( tn) = a+(n-1)d
where a= first term
d= common difference
n = no of terms
so 3rd term (t3)= a+(3-1)d
= a+2d
similarly
7 the term t(7) = a+(7-1)d
=a+6d
6 th term (t6)= a+5d
4 th term (t4) = a+3d
sum of the 3rd and 7th term is 24
so
(a+2d) +(a+6d) = 24
2a+8d =24...................eq(1)
difference between the 6th and 4th is 4
(a+5d)-(a+3d) = 4
2d = 4
divide with 2 on both sides
2d/d = 4/4
d= 2
put d=2 in eq(1)
2a+8*2=24
2a+16=24
2a =24-16
2a= 8
divide with 2 on both sides
+2a%2F2++=+8%2F2
a= 4
so , first term (a) = 4
sum of n terms in Arithmetic series = +%28n%2F2%29%282a%2B%28n-1%29d%29
sum of first 14 terms = +%2814%2F2%29%282%2A4+%2B%2814-1%292%29
=7%288%2B13%2A2%29
=7%288%2B26%29
=7%2834%29
=238
Result : first term = 4 , sum = 238