What number will replace the question mark (?) in the following series?

?, 19, 15, 24, 8, 33

Question

What number will replace the question mark (?) in the following series?

?, 19, 15, 24, 8, 33

Accepted Answer

**Solution:**

1. Observe the series: ?, 19, 15, 24, 8, 33

2. Let's analyze the pattern by checking the differences between consecutive terms starting from the second term:
   - 19 to 15: $15 - 19 = -4$
   - 15 to 24: $24 - 15 = +9$
   - 24 to 8: $8 - 24 = -16$
   - 8 to 33: $33 - 8 = +25$

3. The differences alternate between negative and positive values: -4, +9, -16, +25

4. These numbers are perfect squares with alternating signs:
   -4 = $-2^2$
   +9 = $+3^2$
   -16 = $-4^2$
   +25 = $+5^2$

5. Following this pattern, the difference before 19 should be $-1^2 = -1$

6. Let the missing number be $x$. Then:
   $$19 - x = -1 \ x = 19 + 1 = 20$$

7. However, 20 is not in the options. Let's check if the pattern starts with $-1^2$ or $-2^2$.

8. Alternatively, consider the differences as:
   - From ? to 19: difference = $d_1$
   - From 19 to 15: $-4$
   - From 15 to 24: $+9$
   - From 24 to 8: $-16$
   - From 8 to 33: $+25$

9. The differences are squares of consecutive integers with alternating signs starting from $-4 = -2^2$.

10. So, the difference before 19 should be $+1^2 = +1$ (since the signs alternate: +, -, +, -, +)

11. Therefore:
    $$19 - x = +1 \ x = 19 - 1 = 18$$

12. This matches option (a).

**Answer:** 18

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