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9-92.

Consider this geometric sequence: i0, i1, i2, i3, i4, i5 , …, i15.

  1. You know that i0 = 1, i1 = i, and i2 = −1. Calculate the result for each term up to i15, and describe the pattern.

    i0 = 1 Can you see the pattern?

    i1 = i

    i2 = −1

    i3 = (i)(i2) = −i

    i4 = (i2)(i2) = 1

    i5 = (i2)(i2)(i) = i

    The pattern repeats 1, i, −1, and −i

  2. Use the pattern you found in part (a) to calculate i16, i25, i39, and i100.

    16 is a multiple of 4.
    25 is on more than a multiple of 4.
    39 is one less than a multiple of 4.
    100 is a multiple of 4.

    i16 = 1
    i25 = i
    i39 = −i
    i100 = 1

  3. What is i4n , where n is a positive whole number?

    4n is a multiple of 4.

  4. Based on your answer to part (c), simplify i4n+1, i4n+2, and i4n+3.

    Refer to (a), (b), and (c).

  5. Calculate i396, i397, i398, and i399.

    i396 = i4·99 = 1

    i397 = i4·99+1 = ?

    i398 = i4·99+2 = ?

    i399 = ?