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Question 1027966: The first three terms of a geometric sequence are: T1 ; T2 ; and T3. If T2 = T1 + 4 and T3= T2 + 9, determine the values of T1; T2 and T3
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 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.
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