If the arithmetic mean of \( a_{1}, a_{2}, a_{3}, \ldots, a_{n} \) ...

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If the arithmetic mean of \( a_{1}, a_{2}, a_{3}, \ldots, a_{n} \) is \( a \) and \( b_{1}, b_{2}, b_{3}, \ldots, b_{n} \) have the arithmetic mean \( b \) and
\( \mathrm{P} \) \( a_{i}+b_{i}=1 \) for \( i=1,2,3, \ldots, n \), prove that

W
\[
\sum_{i=1}^{n}\left(a_{i}-a\right)^{2}+\sum_{i=1}^{n} a_{i} b_{i}=n a b
\]
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