if 2-5t;1-t;t+1 are the first 3terms of an infinite converging geometric sequence,determine the values of t
since r is constant,
(1-t)/(2-5t) = (t+1)/(1-t)
Solve that for t, and you will find that matching sequences are, with r=2 and r=1/3,
1/3, 2/3, 4/3, ...
9/2, 3/2, 1/2, ...
To determine the values of "t" in the geometric sequence, we can use the formula for the nth term of a geometric sequence.
The formula for the nth term of a geometric sequence is:
a_n = a * r^(n-1)
Where:
a_n = the nth term
a = the first term
r = the common ratio
n = the position of the term in the sequence
In this case, we are given the first three terms:
a_1 = 2 - 5t
a_2 = 1 - t
a_3 = t + 1
From this information, we can set up a system of equations:
(2 - 5t) = a * r^0 ..... (equation 1)
(1 - t) = a * r^1 ...... (equation 2)
(t + 1) = a * r^2 ...... (equation 3)
Simplifying equation 1, we have:
2 - 5t = a
Substituting this value of "a" into equations 2 and 3:
(1 - t) = (2 - 5t) * r
(t + 1) = (2 - 5t) * r^2
Expanding these equations further:
1 - t = 2r - 5tr
t + 1 = 2r^2 - 5tr^2
Now, we can solve this system of equations to find the value of "t".
From equation 1, we have:
2 - 5t = a
2 - 5t = 1
-5t = -1
t = 1/5
Therefore, the value of "t" in the given geometric sequence is 1/5.
To find the values of "t" for which 2-5t, 1-t and t+1 are the first three terms of an infinite converging geometric sequence, we need to set up an equation based on the definition of a geometric sequence.
In a geometric sequence, each term is obtained by multiplying the previous term by a constant value called the common ratio (r). Therefore, if we denote the first term as "a" and the common ratio as "r," we can express the terms of the sequence as follows:
First term: a = 2-5t
Second term: ar = 1-t
Third term: ar^2 = t+1
To solve for "t," let's set up an equation using the second and third terms:
ar = 1-t (Equation 1)
ar^2 = t+1 (Equation 2)
Now, divide Equation 2 by Equation 1 to eliminate "a":
(ar^2)/(ar) = (t+1)/(1-t)
Cancel out the "r" from the numerator and denominator:
r = (t+1)/(1-t)
To further solve for "t," substitute the value of "r" into Equation 1:
(2-5t)r = 1-t
Replace "r" with (t+1)/(1-t):
(2-5t)(t+1)/(1-t) = 1-t
Now, simplify the equation:
(2-5t)(t+1) = (1-t)(1-t)
Expand and collect like terms:
2t + 2 -5t^2 -5t = 1 - t - t + t^2
Rearrange and simplify:
0 = -7t^2 - 7t + 3
Now, we have a quadratic equation, which we can solve using various methods such as factoring, completing the square, or applying the quadratic formula.
By factoring, we need to find two values that multiply to give +3 and add up to -7. Let's use trial and error to determine the roots:
-1 and -3
So, the factored form of the equation is:
0 = (-t-1)(-t-3)
To solve for "t," set each factor equal to zero:
-t-1 = 0 or -t-3 = 0
Solve each equation separately:
-t-1 = 0 -> -t = 1 -> t = -1
or
-t-3 = 0 -> -t = 3 -> t = -3
Therefore, the values of "t" that satisfy the given conditions are t = -1 and t = -3.