write an algebraic expression that describes this sequence: 0,6,16,30. thanks!
0 when n = 0
*** 6
6 ***** 4
*** 10
16 **** 4
*** 14
30 when n = 3
ah ha, change of change is constant
Of form y = a + b n + c n^2
a = 0 because first term is 0
so of form
y = b n + c n^2
when n = 1, y = 6
6 = b (1) + c (1)
when n = 2, y = 16
16 = b(2) + c (4)
so solve for b and c
12 = 2 b + 2 c
16 = 2 b + 4 c
---------------
- 4 = -2 c
c = 2
then
6 = b + 2 so b = 4
so
y = 4 n + 2 n^2
check
if n = 3
y = 12 + 18 = 30 yes
Write an expression that is equivalent to -15.66 ÷ 5.8. Then, find the quotient.
To write an algebraic expression that describes the given sequence, we need to identify the pattern or relationship between the terms. Let's examine the differences between the terms:
1st difference: 6 - 0 = 6
2nd difference: 16 - 6 = 10
3rd difference: 30 - 16 = 14
The differences between the terms are not constant. However, the second differences are constant. This suggests that the sequence may have a quadratic relationship.
To find the quadratic expression, we will use the general form of a quadratic equation: y = ax^2 + bx + c. Since the second differences are constant, we can set up the equation using the third difference:
14 = a(3^2) + b(3) + c
Simplifying, we have:
9a + 3b + c = 14
Now, let's substitute any term from the sequence into the equation to obtain additional information. Since the first term (0) is substituted into the equation, it will allow us to find the constant term, c.
0 = a(1^2) + b(1) + c
Simplifying, we have:
a + b + c = 0
Now we have a system of equations consisting of:
9a + 3b + c = 14
a + b + c = 0
By solving this system of equations, we can determine the values of a, b, and c. Afterwards, we can write the algebraic expression using those values.