Question 23

Three consecutive positive integers are raised to the first, second and third powers respectively and then added. The sum so obtained is perfect square whose square root equals the total of the three original integers. Which of the following best describes the minimum, say m, of these three integers?


Let us assume that three positive consecutive integers are x, x+1, x+2. They are raised to first, second and third powers respectively. 

$$x^{1} + (x+1)^{2} + (x+2)^{3} = (x + (x+1) +(x+2))^{2}$$

$$x^{1} + (x+1)^{2} + (x+2)^{3}$$ = $$(3x + 3)^{2}$$

$$x^{3} + 7x^{2} + 15x + 9$$ = $$9x^{2} + 9 + 18x$$

After simplifying you get,

$$x^{3} - 2x^{2} - 3x = 0$$

=> x=0,3,-1

Since x is a positive integer, it can only be 3.

So, the minimum of the three integers is 3. Option a) is the correct answer.

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