Evaluate 1+i2+i4+i6+.....+i2n.

Let S = 1 + i^{2} + i^{4} + i^{6} + . . . . . . . + i^{2 n} Let S_{n} = i^{2} + i^{4} + i^{6} + . . . . . . . + i^{2 n} The above sequence is in GP with first term, a = i^{2} and common ratio, r = i^{2} S_{n} = a [\frac{1 - r^{n}}{1 - r}] = i^{2} [\frac{1 - i^{2 n}}{1 - i^{2}}] = \frac{i^{2 n} - 1}{2} Now, S = 1 + \frac{i^{2 n} - 1}{2} = \frac{i^{2 n} + 1}{2}

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