⎩ ⎪ ⎨ ⎪ ⎧ w + 4 x + 9 y + 1 6 z = 1 4 w + 9 x + 1 6 y + 2 5 z = 1 2 9 w + 1 6 x + 2 5 y + 3 6 z = 1 2 3
Let w , x , y , z be real numbers satisfying the system of equations above.
Find 1 6 w + 2 5 x + 3 6 y + 4 9 z .
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Given that
w+4x+9y+16z=1
4w+9x+16y+25z=12
9w+16x+25y+36z=123
To get 16w+25x+36y+49z
=(w+4x+9y+16z) -3(4w+9x+16y+25z) +3(9w+16x+25y+36z)
=1-3(12)+3(123)
=1 - 36 +369
334
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If we see from the pattern at the problem we can make : a n 2 + b ( n + 1 ) 2 + c ( n + 2 ) 2 = ( n + 3 ) 2 solved it we get : a=1, b=-3, c=3. Where a is the first line, b is the second, and c is the third line. Then the answer become 1 × 1 + 1 2 × ( − 3 ) + 1 2 3 × 3 = 3 3 4