Multiple choice

Find ${a}_{30}$ given that the first few terms of a geometric sequence are given by $-2,1,-\dfrac {1}{2},\dfrac {1}{4}....$

  1. $\dfrac {1}{2^{27}}$
  2. $\dfrac {1}{2^{26}}$
  3. $\dfrac {1}{2^{28}}$
  4. $\dfrac {1}{2^{29}}$
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C Correct answer
Explanation

The sequence is -2, 1, -1/2, 1/4. This is a geometric progression with first term a = -2 and common ratio r = -1/2. The n-th term is a * r^(n-1). For n=30, term = -2 * (-1/2)^29 = -2 * (-1 / 2^29) = 2 / 2^29 = 1 / 2^28.

AI explanation

We identify the sequence as a geometric progression with the first term a equal to -2 and the common ratio r equal to -1/2. Using the nth term formula for a geometric progression, Tn = a * r^(n-1), we substitute the values to find the 30th term. This gives T30 = (-2) * (-1/2)^29, which simplifies to (-2/-2^29) or 1/2^28. The result is 1/2^28.

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