## What is the Next Number? 2, 7, 8, 3, 12, 9?

*The given sequence appears to have alternating patterns. First, numbers are increasing by 5 (2 + 5 = 7, 8 + 5 = 13), and then numbers are decreasing by 5 (13 – 5 = 8, 3 – 5 = -2). Following this pattern, the next number should be 12 – 5 = 7. So, the next number in the sequence is 7.*

## Finding Patterns in Number Sequences

Determining the next number in a sequence requires finding a pattern or rule that generates the sequence. Sequences may follow mathematical patterns, increase or decrease by a consistent difference, or be determined based on a complex formula.

Let’s examine the given sequence step-by-step to identify the pattern:

2, 7, 8, 3, 12, 9…

## Examining Differences

Comparing terms, we see:

From 2 to 7 increases by 5 From 7 to 8 increases by 1 From 8 to 3 decreases by 5 From 3 to 12 increases by 9 From 12 to 9 decreases by 3

So the differences bounce around – there’s no constant difference between the terms.

## Testing Operations

We could also try testing different operations on each pair:

7 – 2 = 5 8 / 7 = About 1 3 x 8 = 24 (No match) 12 / 3 = 4 (No match) 9 + 12 = 21 (No match)

This doesn’t reveal an operation linking the sequence either.

## Identifying a Formula

Looking closely, we can spot a formula in the exponents of the terms:

22 = 4 -> 2 72 = 49 -> 7 82 = 64 -> 8 32 = 9 -> 3 122 = 144 -> 12 92 = 81 -> 9

The numbers are the square roots of perfect squares!

## Predicting the Next Term

Using this pattern, the next term would come from:

142 = 196 Square root of 196 is 14

Therefore, the next number in the sequence is 14.

In summary, spotting the squared exponents generating the terms allowed us to identify the underlying rule and predict the next number. Analyzing number patterns builds critical thinking and algebraic reasoning skills.

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