SOLUTION: please help me solve this problem P(x) function polynomial 2nd degree P(1)=1 P(2)=7 P(3)=19 we have natural integer n ⁄ n≥1 proof that P(1)+P(2)+...+P(n)=n^3 please

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: please help me solve this problem P(x) function polynomial 2nd degree P(1)=1 P(2)=7 P(3)=19 we have natural integer n ⁄ n≥1 proof that P(1)+P(2)+...+P(n)=n^3 please      Log On


   

Question 1163528: please help me solve this problem
P(x) function polynomial 2nd degree P(1)=1 P(2)=7 P(3)=19
we have natural integer n ⁄ n≥1
proof that P(1)+P(2)+...+P(n)=n^3
please help me solve this problem
Answer by greenestamps(13195) About Me  (Show Source):
You can put this solution on YOUR website!


1. Finding the quadratic function from the given data points....

A traditional algebraic method....

The general quadratic polynomial is an%5E2%2Bbn%2Bc.

Use the three given data points to form three equations in a, b, and c; then solve the system of equations.

a%2Bb%2Bc+=+1
4a%2B2b%2Bc+=+7
9a%2B3b%2Bc+=+19

3a%2Bb+=+6
5a%2Bb+=+12

2a+=+6
a+=+3

9%2Bb+=+6
b+=+-3

3%2B%28-3%29%2Bc+=+1
c+=+1

ANSWER: The quadratic function for the given points is 3n%5E2-3n%2B1

Here is another method for finding the quadratic function -- far less common than the preceding method; but useful if you know how to use it.

In the quadratic function an%5E2%2Bbn%2Bc, there is a common second difference of 2a.

Use that to determine coefficient a.
    1   7   19   <-- given sequence
      6   12     <-- first differences (differences between successive terms)
        6        <-- second difference (difference between successive first differences)


The second difference is 6; since 2a=6, that means a=3.

Now compare the whole function an%5E2%2Bbn%2Bc to an%5E2 to determine coefficients b and c. The difference between those two functions is the linear polynomial bx%2Bc.

n=1: an^2 = 3(1^2) = 3; P(1) = 1; difference is -2
n=2: an^2 = 3(2^2) = 12; P(2) = 7; difference is -5
n=3: an^2 = 3(3^2) = 27; P(3) = 19; difference is -8

The linear polynomial bx%2Bc that produces the sequence -2, -5, -8 is -3n%2B1.

Therefore, the polynomial that produces the given sequence is %28an%5E2%29 + %28bn%2Bc%29 = %283n%5E2%29 + %28-3n%2B1%29 = 3n%5E2-3n%2B1.

2. Proving that P(1)+P(2)+...+P(n)=n^3....

Prove using mathematical induction. To do that proof, we need to show two things:

(a) P(1) = 1:
P%281%29+=+3%281%5E2%29-3%281%29%2B1+=+3-3%2B1+=+1 OK

(b) The difference between n%5E3 and %28n-1%29%5E3 is P(n).

n%5E3-%28n-1%29%5E3+=+n%5E3-%28n%5E3-3n%5E2%2B3n-1%29+=+3n%5E2-3n%2B1 OK

The proof is complete.