Answer:
a = -2
b = 2
Step-by-step explanation:
The nth term of a sequence is given by the equation:
[tex]u_n=an^2 + bn[/tex]
If the fourth term is -24, then u₄ = -24:
[tex]a(4^2)+b(4)=-24\\\\16a+4b=-24[/tex]
Given the sixth term is -60, then u₆ = -60:
[tex]a(6^2)+b(6)=-60\\\\36a+6b=-60[/tex]
Therefore, we have created a system of equations:
[tex]\begin{cases}16a+4b=-24\\36a+6b=-60\end{cases}[/tex]
To solve the system of equations, begin by rearranging the first equation to isolate b:
[tex]16a+4b=-24\\\\4b=-16a-24\\\\b=-4a-6[/tex]
Substitute b = -4a - 6 into the second equation and solve for a:
[tex]36a+6(-4a-6)=-60\\\\36a-24a-36=-60\\\\12a-36=-60\\\\12a=-24\\\\a=-2[/tex]
Now, substitute the found value of a into the equation for b, and solve for b:
[tex]b=-4(-2)-6\\\\b=8-6\\\\b=2[/tex]
Therefore, the values of a and b are:
[tex]\Large\boxed{\boxed{a = -2}}[/tex]
[tex]\Large\boxed{\boxed{b = 2}}[/tex]
Step-by-step explanation:
we have 2 variables (a, b). so, we need 2 equations with then to solve.
luckily we have been given 2 terms and their sequence numbers. that gives us the 2 equations :
-24 = a×4² + b×4 = 16a + 4b
-60 = a×6² + b×6 = 36a + 6b
based on this the easiest way to solve this is by elimination.
that means we multiply each equation by an individual factor and then add them together. if we chose the factors smartly, the sum eliminates one of the variables, and we can solve for this. and then we can pick any of the original equations to solve for the second variable.
factor for equation 1 : -9/4
(first divide by 4, and then multiply by -9 to get -36a)
factor for equation 2 : 1
-24×-9/4 = 16a×-9/4 + 4b×-9/4
54 = -36a - 9b
-60 = 36a + 6b
------------------------
-6 = 0 - 3b
b = -6/-3 = 2
-24 = 16a + 4b = 16a + 4×2 = 16a + 8
-32 = 16a
a = -32/16 = -2
