Multiple choice

The sum series $i+2+3i+4+ ....$ upto $100$ terms where $i=\sqrt -1$ equals $p+iq$. Then p equals to

  1. 4

  2. 6

  3. 2550

  4. 20
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C Correct answer
Explanation

The series is i + 2 + 3i + 4 + 5i + 6 + ... up to 100 terms. Grouping terms in pairs: (i+2) + (3i+4) + (5i+6) + ... + (99i+100). The real part is 2 + 4 + 6 + ... + 100, which is an arithmetic progression with 50 terms: (50/2) * (2 + 100) = 25 * 102 = 2550.

AI explanation

The series alternates between real numbers and imaginary numbers, where the imaginary terms do not contribute to the real part p. The real terms occur at the second, fourth, and subsequent even positions, forming the sequence 2, 4, 6 up to 100. This is an arithmetic progression with 50 terms, and its sum is calculated as 50 divided by 2 multiplied by 2 plus 100, yielding 2550. Therefore, p equals 2550.

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