Summation of all odd numbers from 1 to 451

The following lines accurately sum when d (common difference) is 1, however, as soon as it changes, I get a wrong answer, does anyone understand why? I assume I am quite ignorant, but I would love any help I can get:

% Initialize n, a, d and s

n = 451;

a = 1;

d = 2;

S = 0;

% Compute S by adding the terms

for i = 1:n

S = S+a+(i-1)*d;

end

%Call S

S

Command Window:

S = 203401

• Where S should actually equal: 51,076

Answer 1

[VBBV]

n = 451;
a = 1;
d = 2;
S = 0;
% Compute S by adding the terms
for i = 1:n
    S(i) = (a+(i-1)*d/2);
end
sum(S(1:2:end)) % sum of all odd numbers 

ans = 51076

[Zacharia]

Thankyou so much, I understand more from both answers now!

I truly appreciate your help, have a lovely day :)

Answer 2

[Voss]

The for loop iterates n times, so after it's done, S represents the sum of n terms, which is fine when d = 1, but when d > 1, the summation you want has less than n terms.

In other words, there are 451 integers between 1 and 451 (inclusive), but only (451+1)/2 = 226 odd integers in that range and only (451+2)/3 = 151 integers congruent to 1 mod 3 (i.e., 1, 4, 7, 10, ...) in that range.

So "for i = 1:n" must be modified to take into account the value of d, so that the loop iterates the correct number of times.

[Zacharia]

That perfectly explains why it only worked for d = 1, I see now.

Thankyou so much for taking time out of you day to explain it, truly helped me

[Voss]

You're welcome!