Multiple choice

If the first three terms of an arithmetic sequence are $a,2a,18a-5b$ and further it is known that $a+b+4=0,$ then the value of the $100th$ term of the sequence would be?

  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 = (18a - 5b) - 2a. This simplifies to a = 16a - 5b, or 15a = 5b, so b = 3a. Substituting into a + b + 4 = 0 gives a + 3a + 4 = 0, so 4a = -4 and a = -1. The sequence is -1, -2, -3, where the n-th term is -n, making the 100th term -100.

AI explanation

In an arithmetic sequence, the common difference is found by subtracting consecutive terms, so d = 2a - a = a. Using the third term, 18a - 5b = 2a + a, which simplifies to 16a - 5b = 0. Solving the system of equations 16a - 5b = 0 and a + b + 4 = 0 yields a = -1 and b = -3. The 100th term is given by a + 99d, which is -1 + 99(-1) = -100.

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