Square ages
Rick's age is a perfect two-digit square and will be so again in 11 years.
How much will the digits of Rick's age add up to a year from now?
The answer is 8.
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1 solution
How do u know it is consecutive
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Since n 2 is the sum of the first n odd numbers (you can see it from the formula below)
∑ k = 1 n 2 k − 1 = 2 ∑ k = 1 n k − ∑ k = 1 n 1 = n ( n + 1 ) − n = n 2
The difference between two squares is the sum of a sequence of consecutive odd numbers.
Here the difference is 1 1 so it must be a sequence with odd lenght. Suppose the lenght was 3 :
2 k + 1 + 2 k + 3 + 2 + 5 = 1 1 ⟺ 6 k = 2
Which has no integer solution. If you suppose the lenght is 5 or higher you would get negative solutions, we can therefore conclude that this two squares are indeed consecutive
The problem states that Rick's age, R, is a perfect square and so is R + 11. So the two consecutive squares whose difference is 11 is 25 and 36. So 1 + 25 = 26 and 2 + 6 = 8 .