Sequence Analysis

We are given the sequence: 48, 24, 8, 5, 5, 20, 11, 6, 3, ?, 9. The goal is to determine the missing number.

Step-by-Step Pattern Derivation

Let's examine the operations connecting consecutive terms to establish the pattern:

• $48 \div 2 = 24$
• $24 \div 3 = 8$
• $8 - 3 = 5$
• $5 \times 1 = 5$
• $5 \times 4 = 20$
• $20 - 9 = 11$
• $11 - 5 = 6$
• $6 \div 2 = 3$

These operations generate the sequence up to the 9th term, which is 3.

Finding the Missing Term (T10)

Let the terms of the sequence be denoted by $T_n$, where $n$ is the position. We need to find $T_{10}$. The sequence provided is $T_1, T_2, ..., T_9, T_{10}, T_{11}$. We know:

• The 5th term, $T_5 = 5$.
• The 9th term, $T_9 = 3$.
• The 11th term, $T_{11} = 9$.

A pattern relating the 10th term ($T_{10}$) to previous terms is observed:

$T_{10} = T_9 + (T_5)^2$

Substitute the known values into this formula:

• $T_{10} = 3 + (5)^2$
• $T_{10} = 3 + 25$
• $T_{10} = 28$

Therefore, the missing number in the sequence is 28.