How can interpolation be used to find a polynomial formula for G(n)?

In summary, the conversation discusses finding a formula to calculate G(n) that should be a polynomial of degree 4. The formula is determined to be G(n) = (1/24)n4 - (1/4)n3 + (11/24)n2 - (1/4)n, which is obtained through the use of the interpolation method and considering known values of G(n). However, the conversation also mentions an incorrect calculation for G(5) and acknowledges a mistake in simple arithmetic.
  • #1
KLscilevothma
322
0
Find a formula to calculate G(n) and it should be a polynomial of degree 4.

G(n) = 1/6(0)(0-1)(0-2) + 1/6(1)(1-1)(1-2)+...+1/6(n-1)(n-2)(n-3)

I know
G(0)=G(1)=G(2)=G(3)=0
G(4)=1
G(5)=1+3
G(6)=1+3+6
G(7)=1+2+3+10
G(8)=1+3+6+10+15 (sum of triangle numbers)

What should I do to find out the forumla? I think it is some kind of interpolation formula.
 
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  • #2
G(n) = SUM[ (1/6)(k-1)(k-2)(k-3) , k=1..n] ..rewrite function
G(n) = (1/6) SUM[ (k-1)(k-2)(k-3) , k=1..n] ..extract constant
G(n) = (1/6) SUM[ (k3-6k2+11k-6) , k=1..n] ..expand the general term on inside
G(n) = (1/6)( SUM[k3, k=1..n] - SUM[6k2, k=1..n] + SUM[11k, k=1..n] - SUM[6, k=1..n] ) ..add sums separately
G(n) = (1/6)( n2(n+1)2/4 - 6n(n+1)(2n+1)/6 +11n(n+1)/2 -6n ) ..rewrite sums
G(n) = (1/24)n4 - (1/4)n3 + (11/24)n2 - (1/4)n ..expand

Your G(n) values do not agree for n > 4. Take for instance G(5) = (1/6)(4-1)(4-2)(4-3) + (1/6)(5-1)(5-2)(5-3) = 1 + 4 = 5
 
  • #3
Originally posted by suffian
G(n) = SUM[ (1/6)(k-1)(k-2)(k-3) , k=1..n] ..rewrite function
G(n) = (1/6) SUM[ (k-1)(k-2)(k-3) , k=1..n] ..extract constant
G(n) = (1/6) SUM[ (k3-6k2+11k-6) , k=1..n] ..expand the general term on inside
G(n) = (1/6)( SUM[k3, k=1..n] - SUM[6k2, k=1..n] + SUM[11k, k=1..n] - SUM[6, k=1..n] ) ..add sums separately
G(n) = (1/6)( n2(n+1)2/4 - 6n(n+1)(2n+1)/6 +11n(n+1)/2 -6n ) ..rewrite sums
G(n) = (1/24)n4 - (1/4)n3 + (11/24)n2 - (1/4)n ..expand

Thank you. I think this is one of the ways to tackle this problem. After reading my notes carefully, I think we can do the question by using "interpolation" method as follows:
G(1)=G(2)=G(3)=0
G(4)=1
G(5)=5
Since G(1)=G(2)=G(3)=0, (x-1), (x-2) and (x-3) are factors of G(n). Since the formula that we need to find is of degree 4, therefore G(n)=(an+b)(n-1)(n-2)(n-3). By using G(4)=1 and G(5)=5, we know G(n)=1/24n(n-1)(n-2)(n-3), which is exactly the answer that you get.

Your G(n) values do not agree for n > 4. Take for instance G(5) = (1/6)(4-1)(4-2)(4-3) + (1/6)(5-1)(5-2)(5-3) = 1 + 4 = 5
Usually I can't do simple arithmatic correctly.
 

What is the interpolation formula?

The interpolation formula is a mathematical equation used to estimate the value of a function at a point that falls between two known data points. It is used to fill in the gaps between data points and is commonly used in data analysis and curve fitting.

What is the purpose of the interpolation formula?

The purpose of the interpolation formula is to provide a method for estimating unknown values based on existing data. This is especially useful when there are missing data points or when the data is not continuous.

How does the interpolation formula work?

The interpolation formula works by finding the equation of a line or curve that passes through two known data points and using that equation to calculate the value at the desired point. The type of interpolation formula used depends on the type of data and the desired level of accuracy.

What are the types of interpolation formula?

There are several types of interpolation formula, including linear interpolation, polynomial interpolation, and spline interpolation. Linear interpolation is the simplest and uses a straight line to connect two data points. Polynomial interpolation uses a polynomial equation to fit a curve through the data points. Spline interpolation uses piecewise polynomial equations to fit a smooth curve through the data points.

What are the limitations of the interpolation formula?

The interpolation formula is limited by the accuracy of the data points and the type of interpolation used. If the data points are not evenly spaced or if there are outliers, the interpolation formula may not accurately estimate the value at the desired point. Additionally, extrapolation (estimating values outside of the given data range) using the interpolation formula is not recommended as it can lead to inaccurate results.

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