# how can i solve equation with summation

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Fatma Abdullah on 30 May 2017
Commented: Rik on 30 May 2017
i can't solve the attached img equation on matlab for : Pfa=10^(-6),M=8,i=s=1
(later i would change these variables)
i tried the following but errors rises
if true
M=8;
s=1;
i=s;
syms pfal M i s
eqn = symsum((nchoosek(M,i)*(pfal)^(i)*(1-pfal)^(M-i)), i, s, M)==8*10^(-6);
solve(eqn,pfal)
end

Rik on 30 May 2017
I think I can read the summation index: it is an i. That would mean that you should initiate i as a sym not as a hard-coded value, and thus the result from nchoosek changes each summation part.
M=8;
s=1;
syms pfal i
f = nchoosek(M,i);
eqn = (symsum(f*(pfal)^(i)*(1-pfal)^(M-i), i, s, M)==10^(-6));
solve(eqn,pfal)
This exits without error, but results in 8 solutions, 6 of which are complex. One solution is indeed equal to the solution that Roger suggested (1.25e-7).
Rik on 30 May 2017
Ah, my bad, I forgot to add the line that converts to double.
result=solve(eqn,pfal);
result=double(result);
result=result( result<1 & abs(imag(result))<2*eps );

Roger Stafford on 30 May 2017
If I interpret your code correctly, you are doing things the hard way. Your expression for Pfa is simply the binomial expansion of ((Pfa1+(1-Pfa1))^M without the first term where i is equal to zero. Hence, you are trying to solve
(Pfa1 + (1-Pfa1))^8 = 1 = 10^(-6) + (1-Pfa1)^8
whose solution is easily found as
Pfa1 = 1 - 0.999999^(1/8)
Probably the cause of your error message is the reference to the numeric function ‘nchoosek’.