looking at the alphabet ,the letters are numbered 1-26 ,
looking at the alphabet ,the letters are numbered 1-26 , such that 1 =one=15+14+5=34 (O=15, N=14, E =5 ) 2=two=20+23+15=58 (T=20, W=23, 0=15) 3=three =56 4=four=60 and so on,
can i find a formula such that given any number i can find their sum with regards to their position in the alphabet?
NUMBER 1 2 3 4 5 6 7 8 9 10 11 12
SUM 34 58 56 60 42 52 65 49 42 39 63 87
$\endgroup$1 Answer
$\begingroup$The answer is yes. Look for Lagrange Interpolation polynomial.
You are looking for a function $f$ such that $f(1),..,f(26)$ takes some particular values.
The Lagrange interpolation sais that no matter what those values are, the following polynomial works:
$$f(1) \frac{(x-2)(x-3)...(x-26)}{(1-2)(1-3)...(1-26)}+f(2) \frac{(x-1)(x-3)...(x-26)}{(2-1)(2-3)...(2-26)}+....+f(26) \frac{(x-1)(x-2)...(x-25)}{(26-1)(26-2)...(26-25)}$$
The formula is a polynomial of degree $25$ or lower (most probably 25), so not really nice...
By adding any function of the form $g(x) (x-1)(x-2)...(x-26)$ you get infinitely many other formulas.
$\endgroup$ 4
