Multiple choice

If the first four terms of an arithmetic sequence are a, 2a, b and a - 6 - b for some numbers 'a' and 'b', then the value of the 100th term is

  1. -100

  2. -300

  3. 150

  4. -150
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A Correct answer
Explanation

In an arithmetic sequence, the difference between consecutive terms is constant. 2a - a = a. So the common difference is a. Third term b = 2a + a = 3a. Fourth term = 3a + a = 4a. Given 4a = a - 6 - b. Substituting b = 3a: 4a = a - 6 - 3a => 4a = -2a - 6 => 6a = -6 => a = -1. Sequence: -1, -2, -3, -4. 100th term = a + (n-1)d = -1 + 99(-1) = -100.

AI explanation

In an arithmetic sequence the difference between consecutive terms is constant, so 2a - a = b - 2a. Solving this gives a = b - 2a, which means b = 3a. The fourth term is a - 6 - b, which becomes a - 6 - 3a = -2a - 6. The common difference from the third term to the fourth term is (-2a - 6) - 3a = -5a - 6. Since the first common difference is a, we have a = -5a - 6, giving 6a = -6 and a = -1. The first term is -1 and the common difference is -1. Using the formula for the nth term of an arithmetic progression, a + (n - 1)d, the 100th term is -1 + 99(-1) = -100.

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