What is the value of X in the sequence 20, 10, 10, 15, 30, 75,
X?

Correct Option

The given sequence is 20, 10, 10, 15, 30, 75, X.

To determine the pattern, the ratio of each term to its preceding term is calculated:

• \(a_2 / a_1 = 10 / 20 = 0.5\)
• \(a_3 / a_2 = 10 / 10 = 1\)
• \(a_4 / a_3 = 15 / 10 = 1.5\)
• \(a_5 / a_4 = 30 / 15 = 2\)
• \(a_6 / a_5 = 75 / 30 = 2.5\)

The sequence of multipliers (0.5, 1, 1.5, 2, 2.5) forms an arithmetic progression with a common difference of 0.5. Following this pattern, the next multiplier for $a_7$ would be \(2.5 + 0.5 = 3\).

Therefore, to find X:

$$X=a_6\times 3$$
$$X=75\times 3$$
\[X = 225\]

Thus, the value of X is 225.

Incorrect Options

Options 1 (105), 2 (120), and 3 (150) are incorrect as they do not conform to the established multiplicative pattern observed in the sequence, where each term is derived by multiplying the preceding term by an incrementing factor of 0.5.