Question

 White  Black  Total +(n+1)2\begin{array} { l r } \text { White } & \text { Black } & \text { Total } \\ & + & ( n + 1 ) ^ { 2 } \end{array}

Solution

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The answer was already given in the previous page:

2n+(n2+1)=n2+12n+(n^2+1) =n^2+1

Using 2n12n-1, the number of white dots correspond to the sequence:

2,4,6,8,2,4,6,8,\dots

Using n2+1n^2+1, the number of black dots correspond to the sequence:

2,5,10,17,2,5,10,17,\dots

which corresponds to sequence A.\text{\textcolor{#c34632}{A.}}

To show that the polynomial addition is true, we notice that n2+2n+1n^2+2n+1 is a perfect square trinomial: a2+2ab+b2=(a+b)2a^2+2ab+b^2=(a+b)^2

2n+n2+1=n2+2n+1=(n+1)2(1)2n+n^2+1= n^2+2n+1=\color{#c34632} (n+1)^2\color{white}\tag{1}

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