LalaB wrote:
I feel that it is just an arithmetic progression with mean=median
It is not an arithmetic progression, so if you assume that it is, you will get the wrong answer. An arithmetic progression is 'equally spaced'. If you just look at the smallest perfect squares, you can see that they are not equally spaced: 1, 4, 9, 16, 25... In fact the spacing gets larger the further you get into this list.
You can use the spacing of perfect squares to answer this question. From the difference of squares, we have that
41^2 - 40^2 = (41 + 40)(41 - 40) = 81
So 41^2 = 40^2 + 81. Similarly, 42^2 = 41^2 + 83, and 39^2 = 40^2 - 79, and so on. Listing all of the values we need to sum:
\(\begin{align*}
36^2 &= 40^2 - 79 - 77 - 75 - 73\\
37^2 &= 40^2 - 79 - 77 - 75 \\
38^2 &= 40^2 - 79 - 77 \\
39^2 &= 40^2 - 79 \\
40^2 &= 40^2 \\
41^2 &= 40^2 + 81 \\
42^2 &= 40^2 + 81 + 83 \\
43^2 &= 40^2 + 81 + 83 + 85 \\
44^2 &= 40^2 + 81 + 83 + 85 + 87
\end{align*}\)
Now adding these in columns, we get (9)(40^2) + 4(81-79) + 3(83 - 77) + 2(85 - 75) + (87 - 73) = 9*1600 + 4*2 + 3*6 + 2*10 + 14 = 14,400 + 8 + 18 + 20 + 14 = 14,460
I'd still probably use the first method outlined in Bunuel's post above, but you can use the spacing of squares to get the answer if you look at the problem in the right way.
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