Maths C1 Sequences and Series help!!

[B333]

The sum of the first six terms of an arithmetic series is 21, and the seventh term is three times the sum of the third and the fourth. Find the first term and the common difference.

 

[T19]

this answer is from http://uk.answers.yahoo.com/question/index?qid=20100926103913AAnj1o9
a1+a2+a3+a4+a5+a6 = 21

a2 = a1+d
a3 = a2+d = a1+2d
a4 = a1+3d
a5 = a1+4d
a6 = a1+5d

21 = 6a1 + 15d
a7 = a1+6d = 7(a3+a4) = 7(a1+2d + a1+3d) = 7(2a1+5d)
a1+6d = 14a1 + 35d

6a1 + 15d = 21
13a1 + 29d = 0 => a1 = -29d/13

6(-29d/13) + 15d = 21
-174d +195d = 273
21d = 273
d = 13

6a1 + 15*13 = 21
a1 = (21-15*13)/6 = -29

The first term, a1 = -29
The common difference, d = 13

Check:
a1 = -29
a2 = -16
a3 = -3
a4 = 10
a5 = 23
a6 = 36
a7 = 49
(-29-16-3+10+23+36) = 21
21 = 21

49 = 7(-3+10)
49 = 7*7
49 = 49

hope it helped

 

[B333]

Thanks. I saw this answer but in this example the seventh term is seven time the sum of the third and the fourth term but in the question above it is only three times the sum.