Lagrange Interpolating Polynomial

A neat way to generate a function that passes through certain points
Wolfram Article
Desmos Demo: shows sequence autocompletion
Desmos Demo 2: shows polynomial that morphs to fit arbitrary point while fitting other points
Desmos Demo 3: shows parabola defined by three moveable points
Desmos Demo 4 shows quartic polynomial defined by four moveable points

$P(x)=\sum _{j=1}^n{P}_j(x),P_j(x)=y_j\prod _{k=1,k\ne j}^n\frac{x-x_k}{x_j-x_k}$

• $P(x)$ is the polynomial
• $n$ is the number of points
• Example: Find the $x$ in this sequence: $33,34,35,x,39$:
• 
• Plugging in 1 gives 33 because all the terms not corresponding to $y_j=33$ are sent to $0$ by $x-x_k$ in the numerator
• The remaining term is still normalized to $1$ by the denominator

Code

In[34]:= Expand[
 InterpolatingPolynomial[{{1, 5}, {2, 10}, {3, 20}, {4, 40}, {5, 
    80}, {6, 160}, {7, b}}, x]]
 
Out[34]= -315 + b + (9353 x)/12 - (49 b x)/20 - (8581 x^2)/12 + (
 203 b x^2)/90 + (15545 x^3)/48 - (49 b x^3)/48 - (3695 x^4)/48 + (
 35 b x^4)/144 + (443 x^5)/48 - (7 b x^5)/240 - (7 x^6)/16 + (
 b x^6)/720
 
In[35]:= Simplify[%]
 
Out[35]= 1/720 (b (720 - 1764 x + 1624 x^2 - 735 x^3 + 175 x^4 - 
      21 x^5 + x^6) - 
   15 (15120 - 37412 x + 34324 x^2 - 15545 x^3 + 3695 x^4 - 443 x^5 + 
      21 x^6))
 
In[33]:= Series[5*2^x,{x,7,9}]
 
Out[33]= SeriesData[x, 7, {
 640, 640 Log[2], 320 Log[2]^2, Rational[320, 3] Log[2]^3, 
  Rational[80, 3] Log[2]^4, Rational[16, 3] Log[2]^5, Rational[8, 9] 
  Log[2]^6, Rational[8, 63] Log[2]^7, Rational[1, 63] Log[2]^8, 
  Rational[1, 567] Log[2]^9}, 0, 10, 1]
 
In[29]:= Simplify[%]
 
Out[29]= SeriesData[x, 0, {5, 
Log[32], Rational[5, 2] Log[2]^2, Rational[5, 6] Log[2]^3}, 0, 4, 1]