Question: What is the value of \( y \) in the given sequence? 2, \( y \), 18, -54, 162, … -6 3 -3 6

What is the value of \( y \) in the given sequence?

2, \( y \), 18, -54, 162, …

-6 3 -3 6

Solution

The sequence is given as \(2, y, 18, -54, 162, \ldots\). Let’s try to identify a pattern in the sequence. First, observe the relationship between the known terms: Start with the third term and the fourth term: \[ \frac{-54}{18} = -3 \] The common ratio is \(-3\). Now verify with the subsequent terms: \[ \frac{162}{-54} = -3 \] It confirms that the sequence is a geometric progression with a common ratio of \(-3\). To find \(y\), which is the second term, use the first term to find \(y\): \[ y = 2 \times (-3) \] \[ y = -6 \] Therefore, the value of \(y\) is \(-6\).