The Power of Three II
If the probability that the absolute value of the difference of any two consecutive squares has the value as of any number in the arithmetic progression , can be expressed as , where and are coprime positive integers, find .
Submit 0 as your answer if you think that the probability is zero.
Try Part I .
The answer is 4.
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1 solution
The difference between any two consecutive squares n , n + 1 can be given as ( n + 1 ) 2 − n 2 = 2 n + 1
Now as we start substituting values for n we get:
1) for n = 1 , d i f f e r e n c e = 3
2) for n = 2 , d i f f e r e n c e = 5
3) for n = 3 , d i f f e r e n c e = 7
As we see here, we get the series of odd numbers, I.e. 3 , 5 , 7 , 9 , 1 1 , 1 3 , 1 5 , …
Now the probability that a number is present in the series mentioned in the question can be found out by checking the above series. We observe that for every 3 numbers in the above series there is only 1 number in the series mentioned in the question.
Hence P = 3 1
1 + 3 = 4