Question 1027966
i got an answer, but the numbers involved are not integers.


here's what i did.


T1 = x
T2 = T1 + 4 = x + 4
T3 = T2 + 9 = (T1 + 4) + 9 = T1 + 13 = x + 13


your 3 numbers are:


x
x + 4
x + 13


you state it's a geometric sequence which means An = A1 * r^(n-1)


this means that:


A1 = A1 * r^(1-1) = A1 * r^0 = A1 * 1 = A1 = x


A2 = A1 * r^(2-1) = A1 * r^1 = A1 * r = x * r


A3 = A1 * r^(3-1) = A1 * r^2 = x * r^2


you wind up with:


A1 = x
A2 = x * r
A3 = x * r^2


A2 is the second term in the sequence which you already know is equal to x + 4


this means that x+4 = x*r


you can solve for r to get r = (x+4)/x


A3 is the third term in the sequence which you already know is equal to x + 13


this means that x+13 = x*r^2


you can solve for r^2 to get r^2 = (x+13)/x


if r = (x+4)/x, then r^2 must be equal to ((x+4)/x)^2 which is equal to (x+4)^2 / x^2


you now have r^2 = (x+4)^2 / x^2 and you have r^2 = (x+13)/x


this means that (x+4)^2 / x^2 = (x+13) / x


multiply both sides of this equation by x^2 and you get:


(x+4)^2 = (x+13) * x


simplify to get:


x^2 + 8x + 16 = x^2 + 13x


subtract x^2 from both sides of the equation and subtract 8x from both sides of the equation and you get:


16 = 5x


solve for x to get x = 16/5


you know that r = (x+4)/x


when x = 16/5, this becomes:


r = (16/5 + 4) / (16/5)


since 4 is equal to 20/5, this equation becomes:


r = (16/5 + 20/5) / (16/5)


simplify to get r = (36/5) / (16/5)


this is the same as r = (36/5) * (5/16) which results in r = 36/16 which simplies to r = 9/4


you have:


x = 16/5
r = 9/4


you were given that:


A1 = x
A2 = x+4
A3 = x+13


when x = 16/5, this becomes:


A1 = 16/5
A2 = 16/5 + 4 = 16/5 + 20/5 = 36/5
A3 = 16/5 + 13 = 16/5 + 65/5 = 81/5


you are also given that:


A1 = x * r^0
A2 = x * r^1
A3 = x * r^2


when x = 16/5 and r = 9/4, this becomes:


A1 = 16/5 * (9/4)^0 = 16/5 * 1 = 16/5
A2 = 16/5 * (9/4)^1 = 16/5 * 9/4 = 144/20 = 36/5
A3 = 16/5 * (9/4)^2 = 16/5 * 81/16 = 1296/80 = 81/5


the geometric formula yields the same answer when x = 16/5 and when r = 9/4.


you were asked to determine the value of T1 and T2 and T3.


those values are:


16/5
36/5
81/5


you were given that T2 = T1 + 4


that would make T2 = 16/5 + 4 = 16/5 + 20/5 = 36/5


you were given that T3 = T2 + 9


that would make T3 = 36/5 + 9 = 36/5 + 45/5 = 81/5


looks like we're good.