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# What is the rule from x to y?

Sun 13 Mar, 2011 08:42 am
x | y
1 | 1
2 | 3
3 | 6
4 | 10
5 | 15
6 | 21
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Type: Question • Score: 2 • Views: 1,542 • Replies: 5
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JPB

1
Sun 13 Mar, 2011 09:06 am
@Rainbowicegirl21,
Y=(x^2+x)/2
Oylok

1
Sun 13 Mar, 2011 10:12 am
@Rainbowicegirl21,
Systematic way of getting to the above answer:

You know that the answer is going to have the form "y = ax^2 + bx + c" by looking at the way that y is increasing.

So a, b and c are just some numbers whose values we don't know. But we know that the statement "y = ax^2 + bx + c" is always true for any of the pairs of numbers in the table.

So by plugging in some of those numbers we can figure out what a, b and c are.

Let x = 1 and y = 1, and the statement gives us 1 = a + b + c. (Equation 1)
Let x = 2 and y = 3, and the statement gives us 3 = 4a + 2b + c. (Equation 2)
Let x = 3 and y = 6, and the statement gives us 6 = 9a + 3b + c. (Equation 3)

Now we have a system of 3 equations and 3 unknowns. You can solve it to figure out what a, b and c are. The answer ends up being a = 1/2, b = 1/2, c = 0.

Thus y = (x^2 + x) / 2.
djjd62

1
Sun 13 Mar, 2011 10:15 am
@Rainbowicegirl21,
x should be preceded by w and y should be followed by z, unless of course you're reciting the alphabet back to front, in which case modify appropriately
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markr

2
Sun 13 Mar, 2011 01:47 pm
@Oylok,
"You know that the answer is going to have the form "y = ax^2 + bx + c" by looking at the way that y is increasing."

For someone who needs help with this kind of problem, that statement is going to seem awfully arbitrary, and it's going to get her into trouble when she tries to apply it to a cubic (or higher degree) function.

She can use finite differences (http://mathworld.wolfram.com/FiniteDifference.html) to determine the degree of the function. In the table below, the first row consists of her original "y" values. The subsequent rows are the differences between successive values of the row above. For example, 3-1=2, 6-3=3, 10-6=4, and 15-10=5.

1 3 6 10 15 21
2 3 4 5 6
1 1 1 1

Since the second row of differences is constant, it is a second order (quadratic) function.
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AM1

1
Mon 21 Mar, 2011 01:38 pm
@JPB,
Hello
We arange the numbers in that way
x:1 2 3 4 5 6
y:1 3 6 10 15 21

We cn write the Y,s in that way
y1=1 = 1
y2= 3 = 1 +2
y3= 6 = 1+2+3
y4=10=1+2+3+4
y5=15=1+2+3+4+5
y6=21=1+2+3+4+5+6
We see that each yn is the sum of the first n numbers from 1 to n

Those n numbers are an arithmetic sequence with 1 as the first

term and 1 is the difference. As you know, tje formula of the sum of

the first n terms of an arithmetic sequence is:

Sn=(2a1+d*(n-1))*n/2

In our case
a1=1
d=1
Yn=Sn=(1+1+1*(n-1))*(n/2)=n*(n+1)/2

Now if x is n and Yn Is a certain Y, the rule is:
Y=x*(x+1)/2

As written in the previous response

I'll happy to answer if there are questions

Amos

amos_s12@zahav.net.il

0 Replies

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