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12 problems match “complex conjugate”. Clear

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11. Term of a complex recurrence ●●●○○ 50 ptsThe sequence of complex numbers $\{z_n\}$ satisfies $z_1=1$ and $z_{n+1}=\overline{z_n}+1+ni$ for $n=1,2,\ldots$. Write $z_{2015}=a+bi$, where $a$ and $b$ are real numbers. Find $a$ and $b$.
12. Complex sequence with conjugate recurrence ●●●○○ 50 ptsThe complex number sequence $\{z_n\}$ satisfies $z_1=1$ and $z_{n+1}=\overline{z_n}+1+ni$ for $n=1,2,3,\dots$, where $i$ is the imaginary unit and $\overline{z_n}$ is the conjugate of $z_n$. Find the real part and the imaginary part of $z_{2015}$.

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