Last Updated: 2 Feb, 2021
Number In AP
Moderate
Asked in companies
You’re given three integers X, C, and Y, where X denotes the first term of an arithmetic sequence whose common difference is C. Your task is to determine whether Y belongs to this arithmetic sequence or not.
For example:
If X is 6 and C is 3, the sequence becomes 6,9,12,15,18,21,24…
Input Format:
The first line of input contains T, the number of test cases.
The first line of each test case contains three space-separated integers X, Y, and C as described in the problem statement.
Output Format:
For each test case print in a new line “True”, if Y belongs to the sequence otherwise “False”.
Note:
You don’t need to print anything. It has already been taken care of. Just implement the given function.
Constraints:
1 <= T <= 10^5
-10^8 <= X,C,Y <= 10^8
Time Limit: 1 sec
Approaches
This is a mathematics question, hence we need to observe a few things and then make cases accordingly to reach our answer.
- Case 1: If C = 0 and X = Y
If the common difference is 0 and both X and Y are equal, this implies that Y is the part of the sequence so we print “True”.
- Case 2: If X > Y and C > 0, or X < Y and C <0
If you observe carefully, you’ll see that if we encounter the above case our answer will always be “False” because if the first term(X) is greater than Y and C is positive then there’s no chance that Y can occur in this sequence because if X > Y, then for Y to be a part of this sequence, C has to be negative and since C > 0, we’ll always get “False”. Similarly if X < Y, then if C is positive then only Y will be a part of the arithmetic sequence, but here C < 0, we’ll get “False”.
- Case 3: If (Y - X) % C = 0 and C is non zero
In this case, we are checking that if the difference between X and Y is a multiple of C or not because if that condition is met, then Y will surely exist in the sequence and our answer will be “True”.
For example : X = 3, C = 3, and Y = 18
Now, (18 - 3 )% 3 = 15 % 3 = 0, this means Y is a part of this arithmetic sequence, which is true because this arithmetic sequence will be 3,6,9,12,15,18,21…. Where 18 is the 6th term of this sequence.
- Case 4: When we don’t encounter any of the above three cases we simply print “False” since there isn't any case where it is possible for Y to be a part of this sequence.