What is the rule for the sequence 20, 17, 13, 8, 2? I've been trying to answer it by second difference method but still don't get correct formula.
20-3=17
17-4=13
13-5=8
8-6=2
the second differences are a constant so the relationship is quadratic.
the the equation be
y = ax^2 + bx + c
when x = 1
a+b+c = 20
when x=2
4a+2b+c = 17
when x = 3
9a+3b+c = 13
subtract 1str from 2nd
3a+b=-3
subtract 2nd from 3rd
5a + b = -4
now subtract those last two:
2a = -1
a = -1/2
in 3a+b=-3
-3/2 + b = -3
b = -3/2
in the 1st
-1/2 - 3/2 + c = 20
c = 22
y or f(x) = (-1/2)x^2 - (3/2)x + 22
or (-x^2 - 3x + 44)/2
To find the rule for the sequence 20, 17, 13, 8, 2, let's use the second difference method.
Step 1: Find the first differences.
The first differences are obtained by subtracting each number from the one before it:
17 - 20 = -3
13 - 17 = -4
8 - 13 = -5
2 - 8 = -6
Step 2: Find the second differences.
The second differences are obtained by subtracting each first difference from the one before it:
-4 - (-3) = -1
-5 - (-4) = -1
-6 - (-5) = -1
Step 3: Check if the second differences are constant.
In this case, the second differences are all -1, which means they are constant.
Step 4: Determine the rule for the sequence.
Since the second differences are constant, we can conclude that the sequence follows a quadratic pattern.
The general formula for a quadratic sequence is:
an = (a × n^2) + (b × n) + c
To find the constants (a, b, and c), we can substitute the values of the known terms into the formula and solve the resulting equations.
Using the first few terms of the sequence:
20 = (a × 1^2) + (b × 1) + c -> Equation 1
17 = (a × 2^2) + (b × 2) + c -> Equation 2
13 = (a × 3^2) + (b × 3) + c -> Equation 3
From these equations, we can solve for a, b, and c.
Subtracting equation 1 from equation 2:
17 - 20 = (a × 2^2) + (b × 2) + c - ((a × 1^2) + (b × 1) + c)
-3 = 4a + 2b - a - b
-3 = 3a + b -> Equation 4
Subtracting equation 2 from equation 3:
13 - 17 = (a × 3^2) + (b × 3) + c - ((a × 2^2) + (b × 2) + c)
-4 = 9a + 3b - 4a - 2b
-4 = 5a + b -> Equation 5
Now, we have two equations (equations 4 and 5) with two unknowns (a and b). We can solve these equations simultaneously to find the values of a and b.
Solving equations 4 and 5, we get:
-3 = 3a + b -> Equation 4
-4 = 5a + b -> Equation 5
Subtracting equation 4 from equation 5:
-4 - (-3) = 5a + b - (3a + b)
-1 = 2a
a = -1/2
Substituting the value of a into equation 4:
-3 = 3(-1/2) + b
-3 = -3/2 + b
b = -3 + 3/2
b = -3/2 + 6/2
b = 3/2
Now, we have the values of a and b.
Finally, substituting the values of a, b, and n into the quadratic formula:
an = (-1/2 × n^2) + (3/2 × n) + c
To find the value of c, we can substitute the first term of the sequence:
20 = (-1/2 × 1^2) + (3/2 × 1) + c
20 = -1/2 + 3/2 + c
20 = 2/2 + c
20 = 1 + c
c = 20 - 1
c = 19
Thus, the rule for the sequence 20, 17, 13, 8, 2 is:
an = (-1/2 × n^2) + (3/2 × n) + 19
To find the rule for the given sequence 20, 17, 13, 8, 2 using the method of second differences, we need to follow these steps:
Step 1: Calculate the first differences by subtracting each term from its successive term.
First differences: -3, -4, -5, -6
Step 2: Calculate the second differences by subtracting each term from its successive term.
Second differences: -1, -1, -1
If the second differences are constant, it suggests that the original sequence is described by a quadratic equation. However, in this case, the second differences are not constant, which means we need to explore a different approach.
An alternative approach is to look for a pattern or possible operations used to generate the sequence. Here's how we can analyze it:
From 20 to 17, we subtracted 3.
From 17 to 13, we subtracted 4.
From 13 to 8, we subtracted 5.
From 8 to 2, we subtracted 6.
By observing the differences in subtraction, it seems that the term number itself was subtracted from the previous term to generate the next term.
Therefore, the rule for the given sequence is:
Start with 20 and then subtract the term number to generate the next term.
For example:
20 - 1 = 19
19 - 2 = 17
17 - 3 = 14
14 - 4 = 10
10 - 5 = 5
5 - 6 = -1
So, the next term in the sequence would be 19.