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There are four arithmetic means between 2 and $- 18$ The means are

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$- 2, - 6, - 10, - 14$

$1, - 4, - 7, - 10$

$- 2, - 5, - 9, - 13$

$- 4, - 7, - 10, - 13$

answer is D.

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Detailed Solution

Let the means be $X_{1}, X_{2}, X_{3}, X_{4}$ and he common difference be b then $2, X_{1}, X_{2}, X_{3}, X_{4}$ , $- 18$ are in AP

$\Rightarrow - 18 = 2 + 5 b \Rightarrow 5 b = - 20 \Rightarrow b = - 4$

Hence, $\begin{array}{l} X_{1} = 2 + b = 2 + (- 4) = - 2; \\ X_{2} = 2 + 2 b = 2 - 8 = - 6 \\ X_{3} = 2 + 3 b = 2 - 12 = - 10; \\ X_{4} = 2 + 4 b = 2 - 16 = - 14 \end{array}$

The required means are $- 2, - 6, - 10, - 14$ .

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