the sequence 3;-2;x;-24...is quadratic sequence. find the value of x
If I understand you correctly, you have the points
(1,3), (2,-2), (3,x), (4,-24)
and those points lie on a quadratic function graph.
let the function by y = ax^2 + bx + c
from (1,3) : a + b + c = 3 --- #1
from (2,-2) : 4a + 2b + c = -2 ---#2
from (4,-24) : 16a + 4b + c = -24 ---#3
#2 - #1: 3a + b = -5
#3 - #1: 15a + 3b = -27 --> 5a + b = -9
subtract those last two:
2a = -4
a = -2
in 3a+b= -5 ---> b = 1
in #1: -2+1+c=3
c = 4
y = -2x^2 + x + 4
so for the missing point (3,x)
x = -18+3+4 = -11
To find the value of x in the quadratic sequence 3, -2, x, -24, we can look for a pattern within the given sequence.
A quadratic sequence is a sequence where the difference between consecutive terms varies linearly. In other words, the second differences between the terms will be constant.
Let's calculate the differences between adjacent terms:
- The difference between -2 and 3 is -2 - 3 = -5.
- The difference between x and -2 is x - (-2) = x + 2.
- The difference between -24 and x is -24 - x = -x - 24.
Now, let's calculate the differences between the differences:
- The difference between -5 and (x + 2) is (x + 2) - (-5) = x + 7.
- The difference between (x + 2) and (-x - 24) is (-x - 24) - (x + 2) = -2x - 26.
Since we are looking for a quadratic sequence, the second differences should be constant. In this case, the second difference is -2x - 26.
To find the value of x, we need to determine when the second difference is equal to zero.
-2x - 26 = 0
Solving this equation:
-2x = 26
x = 26/-2
x = -13
Therefore, the value of x in the quadratic sequence 3, -2, x, -24 is -13.