Solving the Number Series Puzzle

The question asks to find the missing number in the series: 3, 3, 9, 15, 33, ?, 129, 255, 513, 1023.

Identifying the Series Pattern

Let's analyze the relationship between consecutive terms ($a_n$ and $a_{n-1}$) in the given series.

• $a_1 = 3$
• $a_2 = 3$
• $a_3 = 9$
• $a_4 = 15$
• $a_5 = 33$
• $a_6 = ?$
• $a_7 = 129$
• $a_8 = 255$
• $a_9 = 513$
• $a_{10} = 1023$

We can observe a pattern involving multiplication by 2 and alternating addition/subtraction of 3:

• $a_2 = (2 \times a_1) - 3 = (2 \times 3) - 3 = 6 - 3 = 3$
• $a_3 = (2 \times a_2) + 3 = (2 \times 3) + 3 = 6 + 3 = 9$
• $a_4 = (2 \times a_3) - 3 = (2 \times 9) - 3 = 18 - 3 = 15$
• $a_5 = (2 \times a_4) + 3 = (2 \times 15) + 3 = 30 + 3 = 33$

The pattern is: Multiply the previous term by 2 and subtract 3 if the current term index is even, or add 3 if the current term index is odd (starting the check from $a_2$).

General formula:

• $a_n = 2 \times a_{n-1} - 3$, if $n$ is even.
• $a_n = 2 \times a_{n-1} + 3$, if $n$ is odd ($n > 1$).

Calculating the Missing Term ($a_6$)

To find the missing term ($a_6$), we use the pattern with $a_5$ (since 6 is an even index):

• $a_6 = (2 \times a_5) - 3$
• $a_6 = (2 \times 33) - 3$
• $a_6 = 66 - 3$
• $a_6 = 63$

Verifying the Pattern

Let's check if this calculated value fits the rest of the series:

• $a_7 = (2 \times a_6) + 3 = (2 \times 63) + 3 = 126 + 3 = 129$ (Correct)
• $a_8 = (2 \times a_7) - 3 = (2 \times 129) - 3 = 258 - 3 = 255$ (Correct)
• $a_9 = (2 \times a_8) + 3 = (2 \times 255) + 3 = 510 + 3 = 513$ (Correct)
• $a_{10} = (2 \times a_9) - 3 = (2 \times 513) - 3 = 1026 - 3 = 1023$ (Correct)

The pattern holds true for the entire series. The missing number is 63.