Thayer Watkins
Silicon Valley
& Tornado Alley

The Sum of Digits for Polynomials

For polynomials whose coefficients are terminating decimals (this includes integers) the digit sum arithmetic applies. If f(x) is such a polynomial and x is a terminating decimal number then

DigitSum(f(x) = DigitSum(f(DigitSum(x))

For example, if f(x) = x² + x then DigitSum(f(1.1)) = DigitSum(1.32)=6 and f(DigitSum(1.1) = f(2) = 2²+2 = 4 + 2 = 6. Thus DigitSum(f(x) = DigitSum(f(DigitSum(x)).

This property provides a handy check on the accuracy of the computation of the value of a polynomial.

Power Series

A power or geometric series is just a special kind of polynomial so digit sum arithmetic applies. Let x is a terminating decimal number.

S = Σ0nn xj = (xn+1−1)/(x−1)


DigitSum(S) = DigitSum[((DigitSum(x))n+1−1)*Equivalent(DigitSum(x)−1)]
providing that DigitSum(x)−1 is not a multiple of 3

For example, let x=11 and n=6. Then S=(117−1)/(11−1)=1948717 and so DigitSum(S)=1. DigitSum(x)=2 and DigitSum(27−1)=DigitSum(127)=1. Since DigitSum(x)−1=1 the value computed is equal to the DigitSum(S).

The arithmetic can be extended to other series for which there are formulas.

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