In the number series 2, 3, 5, 12, 50, 408, 6434, find the wrong number.

Steps to Solve

1. Identify the pattern

Looking at the series, we can observe a pattern where each number is derived from the previous two numbers. The pattern seems to be: $a_n = a_{n-1} * a_{n-2} - 1$.

2. Check the pattern with the first few numbers

Let's verify this pattern:

• $5 = 3 * 2 - 1$
• $12 = 5 * 3 - 3$ (should be $5 * 3 - 1 = 14$, which indicates an error)
• $50 = 12 * 5 - 10$ (using the original $12$, which is already suspected to be wrong)

3. Correct the series assuming 12 is wrong and re-calculate

Let's call the correct version of 12, $x$. Then the series will look like 2, 3, 5, $x$, 50, 408, 6434. We can see that $x = 5 * 3 - 1 = 14$, so let's replace 12 with 14 and continue: 2, 3, 5, 14, 50, 408, 6434.

4. Continue the series to see whether the next numbers match up Let's check $50 = 14 * 5 - y$, where y may not be 1 $50 = 70 - y$ $y = 20$ It seems that the original formula $a_n = a_{n-1} * a_{n-2} - 1$ does not work. Let's try another one.

5. Identify the correct pattern The correct pattern is $a_n = a_{n-1} * a_{n-2} - a_{n-3}$.

6. Checking the pattern $5 = 3 * 2 - 1 = 5$ (Here 1 is the number from 3 steps back, which corresponds to number 2) $12 = 5 * 3 - 2 = 13$ (12 is not correct and should be 13. Here 2 is the number from 3 steps back) $50 = 12 * 5 - 3 = 57$ (50 is not correct. Here 3 is the number from 3 steps back)

7. Find another possible pattern

The pattern is $x_n = x_{n-1} + (x_{n-2} * 2)$.

8. Checking this pattern $5 = 3 + (2 * 1) = 5$. (Here 1 is not 2) $12 = 5 + (3 * 2) = 11$. (12 is incorrect) So the pattern is $x_n = x_{n-1} + (x_{n-2} * 2)$ If 12 were 11, then $50 = 11 + (5 * 2) = 21$ (50 is incorrect)

9. Identify the correct pattern The pattern is $x_n = (x_{n-1} * x_{n-2}) / 2$ rounded to the nearest integer.

10. Check the pattern and identify the incorrect number $x_3 = (3 * 2) / 2 = 3$ which is close to 5. $x_4 = (5 * 3) / 2 = 7.5 \approx 8$. 12 is incorrect here. So the sequence should be 2, 3, 5, 8... Then $x_5 = (8 * 5) / 2 = 20$, very different from 50. Let's assume 50 is correct and 12 is incorrect: 2, 3, 5, x, 50... Then we replace 12 with x: $x_5 = (x * 5) / 2 = 50$ $x * 5 = 100$ $x = 20$ So the sequence could be 2, 3, 5, 20, 50... Let's check what 408 would be: $x_6 = (50 * 20) / 2 = 500$. 408 is the incorrect number.

11. Conclude

From the above, it appears that 408 is the incorrect number.