Simplify (i+1)^{3200}-(i-1)^{3200}
= 2^1600 - 2^1600 =0 {Per Mathematica 11] !!!!.
+129852 (i + 1)^3200 =
i^3200 + ai^3199 + bi^3198 + ci^3197 + di^3196 + .....+ di^4 + ci^3 + bi ^2 + ai + 1
(i - 1)^3200 =
i^3200 - ai^3199 + bi^3198 - ci^3197 + di^3196 - .....+ di^4 - ci^3 + bi ^2 - ai + 1
So
(i + 1)^3200 - (i - 1)^3200 leaves
2 [ ai^3199 + ci^3197 + ......+ ci^3 + ai ] =
2 [ a (-i + i) + c (i + - i) + e( -i + i) + g ( i + - i) + ....... ] =
2 [ a * 0 + c * 0 + e * 0 + g * 0 + ...... ] =
2 [ 0 ] =
0
+26393 Simplify (i+1)^{3200}-(i-1)^{3200}
\(\begin{array}{|rclrcl|} \hline && \mathbf{(i+1)^{3200}-(i-1)^{3200}} \\ &=& (i+1)^{2\cdot 1600}-(i-1)^{2\cdot 1600} \\ &=& \left( (i+1)^{2} \right)^{1600}- \left( (i-1)^{2} \right)^{1600} \quad & |\quad (i+1)^{2} &=& i^2+2i+1 \qquad i^2 = -1\\ && \quad & \quad &=& -1+2i+1 \\ && \quad & \quad &=& 2i \\ &=& (2i)^{1600} - \left( (i-1)^{2} \right)^{1600} \quad & |\quad (i-1)^{2} &=& i^2-2i+1 \qquad i^2 = -1\\ && \quad & \quad &=& -1-2i+1 \\ && \quad & \quad &=& -2i \\ &=& (2i)^{1600} - (-2i)^{1600} \\ &=& (2i)^{1600} - (-1)^{1600}(2i)^{1600} \quad & \quad (-1)^{1600} = 1 \\ &=& (2i)^{1600} - (2i)^{1600} \\ &\mathbf{=}& \mathbf{0} \\ \hline \end{array}\)
