Consecutive Products
The integers a , b , c , d are consecutive integers in an increasing order respectively. If b c = 5 5 2 , what is the value of a d ?
The answer is 550.
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1 solution
I suppose the consecutive sequence − 2 5 , − 2 4 , − 2 3 , − 2 2 would also work.
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Yes, that's why I'd rather put "applicable" to "solution" sequence in the comment.
I think you meant "consecutive integers".
Plus, you still need to show there exists 4 consecutive integers that satisfy this condition.
I got it wrong by sillily assuming the four terms ( a − 2 ) , ( a − 1 ) , ( a + 1 ) , ( a + 2 ) with common difference 2. But anyway here is the follow up for that:
- Let the numbers in A.P. be: ( a − 1 . 5 ) , ( a − 0 . 5 ) , ( a + 0 . 5 ) , ( a + 1 . 5 ) .
- We are given ( a − 0 . 5 ) ( a + 0 . 5 ) = 5 5 2 ⟹ a 2 = 5 5 2 . 5 .
- Then: ( a − 1 . 5 ) ( a + 1 . 5 ) = a 2 − 2 . 2 5 ⟹ ( 5 5 2 . 5 ) − 2 . 2 5 = 5 5 0 .
Since x ( x + 1 ) = x 2 + x = 5 5 2 , ( x − 1 ) ( x + 2 ) = x 2 + x − 2 = 5 5 2 − 2 = 5 5 0 .
As such the applicable consecutive sequence could be 2 2 , 2 3 , 2 4 , 2 5 .