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A polynomial $P(x)$ of $n$ degree satisfies $P(k)=2^k$ for $k = 0,1,2,3......,n$. Find the value of $P(n+1)$.

How can I proceed in solving such problems.

Blaise Thunderstorm
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  • You have degree $n$ and $n+1$ particular values. So set it all up with $n+1$ arbitrary coefficients and plug in. – coffeemath Dec 12 '16 at 12:27
  • Just determine the interpolating polynomial. I do not think that there is any shortcut here – Peter Dec 12 '16 at 12:30
  • The other two comments suggest the obvious brute-force approach. There might be some clever approach using forward differencing. Not sure; not feeling very clever tonight. – bubba Dec 12 '16 at 12:35
  • Another possible way: Use Lagrange-Polynomials https://en.wikipedia.org/wiki/Lagrange_polynomial – Dominik Dec 12 '16 at 12:46

1 Answers1

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$P_n(x+1) - P_n(x) = P_{n-1}(x)$

$G(n) = P_n(n+1)$

$G_n = P_n(n+1) - P_n(n) + P_n(n) = 2^n + G_{n-1}$

So, is the simple close formula

kotomord
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