### Will the number 777 be in the sequence that increase by 7 and why?

That all depends upon the first term of the sequence: As long as the first number is less than, or equal to 777 and as long as (first_number MOD 7) ≡ (777 MOD 7) will 777 be in the sequence. 777 MOD 7 ≡ 0 ⇒ if, and only if, first_number ≤ 777 and first_number MOD 7 ≡ 0 (ie 7 divides the first number) will 777 be in the sequence.