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 an t3
t1
r t1 = t1-4
r^2 t1 = (t1-4) -9 = t1 - 13
r (t1-4) = t1-13
r = (t1-13)/(t1-4) = (t1-4)/t1
so
t1^2 - 13 t1 = t1^2 -8 t1 + 16
-5 t1 = 16
t1 = -16/5 = -3.2
then r = (t1-4)/t1
= (-16/5 - 20/5)/-16/5 = 36/16 = 9/4
t2 = -16/5*9/4 = -7.2 = of course
t3 = -7.2*9/4 = -16.2 indeed -7.2-9
To determine the values of t1, t2, and t3 in the given geometric sequence, we can use the information provided.
We are given that t2 = t1 - 4 and t3 = t2 - 9.
Since a geometric sequence has a common ratio between any two consecutive terms, we can write:
t2 = t1 * r
t3 = t2 * r
Substituting t2 = t1 - 4 into the first equation:
t1 - 4 = t1 * r
Substituting t3 = t2 - 9 and t2 = t1 - 4 into the second equation:
t1 - 13 = (t1 - 4) * r
Now we have a set of two equations. We can solve them simultaneously to find the values of t1, t2, and t3.
Let's solve the equations step by step:
Equation 1: t1 - 4 = t1 * r
Rearranging the equation:
t1*r - t1 = -4
t1*(r - 1) = -4
t1 = -4 / (r - 1) ---- (equation 1)
Equation 2: t1 - 13 = (t1 - 4) * r
Expanding and rearranging the equation:
t1 - 13 = t1*r - 4*r - 4
t1*r - t1 + 4*r = -9
t1*(r - 1) + 4*r = -9
Using the value of t1 from equation 1:
(-4 / (r - 1))*(r - 1) + 4*r = -9
Simplifying:
-4 + 4*r = -9
4*r = -9 + 4
4*r = -5
r = -5/4 ---- (equation 2)
Now we have the value of r. We can substitute it back into equation 1 to find t1:
t1 = -4 / (r - 1)
Substituting r = -5/4:
t1 = -4 / (-5/4 - 1)
t1 = -4 / (-5/4 - 4/4)
t1 = -4 / (-9/4)
t1 = -4 * (-4/9)
t1 = 16/9
Therefore, t1 = 16/9.
Now we can substitute t1 back into the equation t2 = t1 - 4 to find t2:
t2 = t1 - 4
t2 = 16/9 - 4
t2 = 16/9 - 36/9
t2 = -20/9
Therefore, t2 = -20/9.
Finally, we can substitute t2 back into the equation t3 = t2 - 9 to find t3:
t3 = t2 - 9
t3 = -20/9 - 9
t3 = -20/9 - 81/9
t3 = -101/9
Therefore, t3 = -101/9.
In summary, the values of t1, t2, and t3 are:
t1 = 16/9
t2 = -20/9
t3 = -101/9