Multiple choice

The sum of first 100 terms of the series 12 - 22 + 32 - 42 + ......... is

  1. -5050

  2. -4950

  3. -1101

  4. 240

  5. -5000
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A Correct answer
Explanation

The series is (1^2 - 2^2) + (3^2 - 4^2) + ... + (99^2 - 100^2). Each pair (n^2 - (n+1)^2) equals -(2n+1). There are 50 such pairs, so the sum is -(3 + 7 + 11 + ... + 199). This is an arithmetic progression with 50 terms, sum = (50/2) * (3 + 199) = 25 * 202 = 5050. Thus, the total is -5050.

AI explanation

When paired together, the first 100 terms of the series form 50 pairs of the form (12 - 22), (32 - 42), and so on. Using the difference of squares formula, each pair equals a difference of negative 10. Multiplying this 50 pairs by negative 10 gives a total sum of negative 5050.

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