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Simplify powers and sums of powers of i

This type of question, asked 4 times: 9 May 2024 S1, 19 Apr 2026 S2, 20 Apr 2026 S2, 14 May 2026 S2. All 4 questions below, oldest first.

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  1. Question 1

    9 May 2024 S1

    110
    If Z1=4i40−5i35+6i17+2Z_1=4i^{40}-5i^{35}+6i^{17}+2, Z2=−1+iZ_2=-1+i, where i=−1i=\sqrt{-1}, then ∣Z1+Z2∣=|Z_1+Z_2|=

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  2. Question 2

    19 Apr 2026 S2

    51
    If z=∑n=02026inz = \sum_{n=0}^{2026} i^n, where i=−1i = \sqrt{-1}, then one of the values of z\sqrt{z} is...

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  3. Question 3

    20 Apr 2026 S2

    52
    If i=−1i = \sqrt{-1} then [i18+(1i)25]3=\left[i^{18} + \left(\dfrac{1}{i}\right)^{25}\right]^3 =

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  4. Question 4

    14 May 2026 S2

    54
    The value of ∑n=14(in+in+1)\displaystyle\sum_{n=1}^{4}\left(i^n + i^{n+1}\right) is

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