Series 5 4 2 -2 -10 Pattern Analysis

The question asks us to identify the next number in the given numerical series: 5, 4, 2, -2, -10, ? We need to find the underlying pattern to determine the missing term.

Finding the Pattern

Let's examine the relationship between consecutive terms by calculating the differences:

• 1st to 2nd term: $4 - 5 = -1$
• 2nd to 3rd term: $2 - 4 = -2$
• 3rd to 4th term: $-2 - 2 = -4$
• 4th to 5th term: $-10 - (-2) = -10 + 2 = -8$

The differences are -1, -2, -4, -8. We can observe a clear pattern here: each difference is twice the value of the preceding difference, and all are negative.

Specifically, the differences form a geometric progression where each term is multiplied by 2 to get the next: $-1 \times 2 = -2$, $-2 \times 2 = -4$, $-4 \times 2 = -8$. This sequence of differences can be described by the formula $-1 \times 2^{n-1}$ for the $n$-th difference in the sequence.

Calculating the Next Term

To find the next term in the series, we first need to determine the next difference. Following the established pattern, the next difference should be:

Next difference = $-8 \times 2 = -16$.

Alternatively, using the formula, the 5th difference (which follows the 4th difference of -8) is calculated as $-1 \times 2^{5-1} = -1 \times 2^4 = -1 \times 16 = -16$.

Now, we add this calculated next difference to the last known term of the series (which is -10) to find the missing term:

Missing Term = Last Term + Next Difference

Missing Term = $-10 + (-16)$

Missing Term = $-10 - 16$

Missing Term = -26

Result

Therefore, the number that should come in place of the question mark (?) in the given series is -26.