4m + 1, m + 1 and 10 - m are the first three terms of a geometric sequence. Find t4.
since the difference is constant,
(m+1)-(4m+1) = (10-m)-(m+1)
m = -9
so the sequence is
-35, -8, 19, 46, ...
But they are not geometric sequence..
To find the fourth term (t4) of a geometric sequence, we need to determine the common ratio (r) and then use it to calculate t4.
In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio (r).
Given the first three terms of the sequence: 4m + 1, m + 1, and 10 - m, we can form two equations by comparing the terms:
Second term / First term = Third term / Second term
(m + 1) / (4m + 1) = (10 - m) / (m + 1)
To solve this equation, we can cross-multiply:
(m + 1) * (m + 1) = (4m + 1) * (10 - m)
Expanding both sides gives:
m^2 + 2m + 1 = 40 - 5m + 10m - m^2
Combine like terms and move everything to one side:
2m^2 + 7m - 39 = 0
Now we can solve this quadratic equation using methods like factoring, completing the square, or the quadratic formula. Let's use the quadratic formula:
m = (-b ± √(b^2 - 4ac)) / (2a)
Here, a = 2, b = 7, and c = -39:
m = (-7 ± √(7^2 - 4 * 2 * -39)) / (2 * 2)
m = (-7 ± √(49 - (-312))) / 4
m = (-7 ± √(49 + 312)) / 4
m = (-7 ± √361) / 4
m = (-7 ± 19) / 4
This gives us two possible values for m:
m1 = (-7 + 19) / 4 = 3
m2 = (-7 - 19) / 4 = -6
Now that we have the possible values for m, we can find the common ratio (r) by substituting each value into the equation:
r = (m + 1) / (4m + 1)
For m1 = 3:
r1 = (3 + 1) / (4 * 3 + 1) = 4 / 13
For m2 = -6:
r2 = (-6 + 1) / (4 * -6 + 1) = -5 / -23 = 5 / 23
Now we can calculate the fourth term (t4) using the common ratios we found:
For m1 = 3:
t4 = (4m + 1) * r1^3 = (4 * 3 + 1) * (4 / 13)^3 = 13 * (4 / 13)^3 = 13 * (64 / 2197) = 832 / 2197
For m2 = -6:
t4 = (4m + 1) * r2^3 = (4 * -6 + 1) * (5 / 23)^3 = (-23) * (5 / 23)^3 = -125 / 23
So, the fourth term (t4) of the geometric sequence is either 832 / 2197 or -125 / 23, depending on the value of m.