hard math problems! please help?

Question

find the rule for the following problems:

1) In: 1 2 3
Out:11 18 27


2) In: 1 2 3
Out: 9 16 25


3) In: 1 2 3
Out: 13 22 33

please help! i will give you 10 points!

ive been spending 2 stupid hours on those 3 problems >:O

Answer 1

Helmut has given you the correct answers but not explained why.

The idea here is to use finite differences. I will explain the first problem and leave the remaining two as an exercise for you.

First construct a finite difference tree:


------1st Diff------2nd Diff

6
---------5
11----------------------2
---------7
18----------------------2
---------9
27----------------------2
--------11
38

You will notice that the 3rd order differences are zero. You can find the general sum of the 1st order differences as follows:

5*nC1 + 2*nC2

where nC1 is the combination of n things taken 1 at a time, nC2 is the combination of n things taken 2 at a time. Thus we have:
5*n+n(n-1)
since 5*nC1 = 5*n and 2*nC2 = n(n-1)
So,
5n+n^2-n = n^2+4n
Since this is the sum to n terms and we have to adjust this by a difference of the first term, we examine our tree and see that this first term is 6. So the sum plus 6 will provide a general formula for finding the nth term. i.e.

n^2+4n+6

So let's check:
n=1 => 1+4+6= 11
n=2 => 4+8+6= 18
n=3 => 9+12+6= 27

Similarly, for problem 2 you have the same sum but a different first term, i.e. 4 so that you obtain: n^2+4n+4 or (n+2)^2

For problem 3, you evaluate 7*nC1+2*nC2 and add 6 to obtain the general formula.

Of course you can also use the general formula for an arithmetic sum rather than the above method if you are still in high school. However, the above method is worth learning because you can derive almost any general formula for sums including squares, cubes, etc.

F.Y.I: James Gregory and Isaac Newton developed differential calculus directly from the study of finite differences. In fact, any derivative and integral can be expressed directly in terms of finite differences. An interesting website is:

http://www.geocities.com/john_gabriel/mlinkexamples.html

Answer 2

This probably not the expected method, and might be above your current level..

See it as you are given 3 points on a cartesian coordinate system and asked to find a curve that joins them. Any polynomial of degree 2 or above will surely able to join these points.

Thus, analyze the differences:
E.g. in 1 are 7 and 9
Thus the differences increase by 2 starting from 7.
Formulate an expression where rate of increment of gradient is 2.
f''(x) = 2
f'(x) = 2x + c1
f(x) = x^2 + c1x + c2
When x = 1, 1^2 + c1 + c2 = 11
When x = 2, 2^2 + 2c1 + c2 = 18
Thus, c1 = (18-11) - (4 - 1) = 4
and c2 = 11 - 1 - 4 = 6
So, a rule is x^2 + 4x + 6.
Check: when x = 3, 3^2 + 4(3) + 6 = 9 + 12 + 6 = 27

Answer 3

Here the pattern seems to be, adding 2 to the last addition
11+7=18, 18+9=27, hence the next will be 27+11=36
9+7=16, 16+9=25, hence 25+11=36
13+9=22, 22+11=33, hence 33=13=46

Answer 4

1. Let S be a set such that for each element x of S there exists a unique element x' of S. 2. There is an element in S, we shall call it 1, such that for every element x of S, 1 is not equal to x'. 3. If x and y are elements of S such that x' = y', then x = y. 4. If M is any subset of S such that 1 is an element of M, and for every element x of M, the element x' is also an element of M, then M = S. Just as a matter of notation, we write 1' = 2, 2' = 3, etc. We define addition in S as follows: (a1) x + 1 = x' (a2) x + y' = (x + y)' The element x + y is called the sum of x and y. Now to prove that 1 + 1 = 2. From (a1), with x = 1, we see that 1 + 1 = 1' = 2.

Answer 5

1)
2) the rule is (2+a)^2

Answer 6

1. If n is the input variable, then n^2 + 4n + 6 is the output

2. If n is the input variable, then (n+2)^2 is the output

3. If n is the input variable, then n^2 + 6n + 6 is the output

Answer 7

1)
N = 6 + 4n + n^2 = 6 +n(4 + n)

2) N = (n + 2)^2

3)
N = 6 + 6x + x^2 = 6 + n(6 + n)

Answer 8

The rule for number 2 is you add 2 to the number you put in and square it to get the number out.

Answer 9

the answer is get off the computer and concentrate on your homework.

Answer 10

wwow, too mind boggling for me