I have a Q that im trying to solve:

it says: the 1st , 2nd, and 3rd term of a geo. sequence is the same as the 1st, 7th and 9th terms of an arith. sequence.: I have to find the common ratio.

So i know: a+8d = a(r^2) (1)
a+6d = ar (2)

but i don't know how to solve this simulataneously! if I divide 1 with 2 i get r = (a+8d) / (a+6d) don't knwo how I should solve it.. iknow the answer is (1/3) in the book. please help.

ok, let's call your first result of

r = (a+8d) / (a+6d) #1 equation

You could also subtract the two equations to get
ar^2 - ar = 2d
r^2 - r = 2d/a let's call that #2

Here is where it gets messy...
sub #1 into #2
(a+8d)^2/(a+6d)^2 - (a+8d)/(a+6d) = 2d/a
finding a common denominator of (a+6d)^2 and simpifying the left side, I got
(2ad + 16d^2)/(a+6d)^2 = 2d/a
divide both sides by 2d

(a+8d)/(a+6d)^2 = 1/a
Cross-multiply and simplify to get
a^2 + 12ad + 3d^2 = a^2 + 8ad
36d^2 + 4ad = 0
9d^2 + ad = 0
d(9d + a) = 0
d = 0 (the arithmetic sequence would not change) OR
d = -9a
back into #1
r = (a+8d)/(a+6d)
= (-9d + 8d)/(-9d + 6d)
= -d/-3d
= 1/3

To solve the simultaneous equations, you can use the elimination method. Here's how you can proceed:

First, expand equation (1) to eliminate the brackets:
a + 8d = ar^2

Next, rearrange equation (2) to isolate 'ar' on one side:
a + 6d = ar

Now, multiply equation (2) by r to match the terms:
(ar)r = (ar)(a+6d)
ar^2 = a^2r + 6adr

Since we already know that a + 8d = ar^2 from equation (1), we can substitute this into the above equation:
a + 8d = a^2r + 6adr

Rearrange this new equation to isolate 'd':
8d - 6adr = a - a^2r

Now, factor out 'd' on the left side:
d(8 - 6ar) = a - a^2r

Finally, solve for 'r' by dividing both sides by 'd' and simplifying the equation:
8 - 6ar = (a - a^2r) / d
8 - 6ar = a/d - a^2r/d
8 = a/d - a^2r/d + 6ar
8 = (a - a^2r + 6ar) / d

Since the first, second, and third terms of the geometric sequence are the same as the first, seventh, and ninth terms of the arithmetic sequence, we can equate them:

a = a
a + d = a + 6d
a + 2d = a + 8d

Now, substitute the values of a, a + d, and a + 2d into the equation we derived:
8 = (a - a^2r + 6ar) / d

Since these terms are equal, we can simplify the equation:
8 = (0 - a^2r + 6ar) / d
8 = (6ar - a^2r) / d
8d = 6ar - a^2r

Now, substitute the values from equation (1) into this equation:
8d = 6a(r^2) - a^2r

Rearrange this equation to isolate 'r':
8d = 6a(r^2) - a^2r
8d = ar^2 - a^2r
0 = ar^2 - a^2r - 8d

Now, you can solve this quadratic equation for 'r'. You can use factoring, completing the square, or the quadratic formula to find its solutions:
0 = r^2 - (a/d)r - 8d/(-a)

Once you have the solutions for 'r', you can determine the common ratio by selecting the appropriate value that satisfies the condition stated in the question. In this case, the common ratio is 1/3, as mentioned in the book.