This worksheet helps students practice identifying consecutive number sequences by working backward from their sums—a fundamental algebra skill.

Let's solve each problem step by step. We'll use algebra to find the consecutive numbers.

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Understanding Consecutive Numbers

- Consecutive numbers are numbers that follow one after another in order.
- For example: 25, 26, 27, 28
- Consecutive even or odd numbers increase by 2 (e.g., 2, 4, 6 or 3, 5, 7)

But here, unless specified otherwise, we assume consecutive integers (increasing by 1).

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General Strategy

For n consecutive numbers, let the first number be $ x $. Then:
- 3 consecutive numbers: $ x, x+1, x+2 $
- 4 consecutive numbers: $ x, x+1, x+2, x+3 $

We sum them and set equal to the given total, then solve for $ x $.

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## ✔ Problems

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1. The sum of 3 consecutive numbers is 39. What are the numbers?

Let the numbers be:
$ x, x+1, x+2 $

Sum:
$ x + (x+1) + (x+2) = 39 $
$ 3x + 3 = 39 $
$ 3x = 36 $
$ x = 12 $

So the numbers are:
12, 13, 14

✔ Check: $ 12 + 13 + 14 = 39 $ ✔️

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2. The sum of 3 consecutive numbers is 78. What are the numbers?

$ x + (x+1) + (x+2) = 78 $
$ 3x + 3 = 78 $
$ 3x = 75 $
$ x = 25 $

Numbers: 25, 26, 27

✔ Check: $ 25 + 26 + 27 = 78 $ ✔️

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3. The sum of 3 consecutive numbers is 129. What are the numbers?

$ 3x + 3 = 129 $
$ 3x = 126 $
$ x = 42 $

Numbers: 42, 43, 44

✔ Check: $ 42 + 43 + 44 = 129 $ ✔️

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4. The sum of 3 consecutive numbers is 60. What are the numbers?

$ 3x + 3 = 60 $
$ 3x = 57 $
$ x = 19 $

Numbers: 19, 20, 21

✔ Check: $ 19 + 20 + 21 = 60 $ ✔️

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5. The sum of 3 consecutive numbers is 372. What are the numbers?

$ 3x + 3 = 372 $
$ 3x = 369 $
$ x = 123 $

Numbers: 123, 124, 125

✔ Check: $ 123 + 124 + 125 = 372 $ ✔️

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6. The sum of 3 consecutive numbers is 300. What are the numbers?

$ 3x + 3 = 300 $
$ 3x = 297 $
$ x = 99 $

Numbers: 99, 100, 101

✔ Check: $ 99 + 100 + 101 = 300 $ ✔️

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7. The sum of 3 consecutive numbers is 273. What are the numbers?

$ 3x + 3 = 273 $
$ 3x = 270 $
$ x = 90 $

Numbers: 90, 91, 92

✔ Check: $ 90 + 91 + 92 = 273 $ ✔️

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Challenge Problems

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1. The sum of 4 consecutive numbers is 98. What are the numbers?

Let numbers be: $ x, x+1, x+2, x+3 $

Sum:
$ x + (x+1) + (x+2) + (x+3) = 98 $
$ 4x + 6 = 98 $
$ 4x = 92 $
$ x = 23 $

Numbers: 23, 24, 25, 26

✔ Check: $ 23 + 24 + 25 + 26 = 98 $ ✔️

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2. The sum of 4 consecutive numbers is 126. What are the numbers?

$ 4x + 6 = 126 $
$ 4x = 120 $
$ x = 30 $

Numbers: 30, 31, 32, 33

✔ Check: $ 30 + 31 + 32 + 33 = 126 $ ✔️

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## ✔ Final Answers:

| Problem | Numbers |
|--------|---------|
| 1 | 12, 13, 14 |
| 2 | 25, 26, 27 |
| 3 | 42, 43, 44 |
| 4 | 19, 20, 21 |
| 5 | 123, 124, 125 |
| 6 | 99, 100, 101 |
| 7 | 90, 91, 92 |
| Challenge 1 | 23, 24, 25, 26 |
| Challenge 2 | 30, 31, 32, 33 |

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🔍 Key Insight:

- For 3 consecutive numbers: Sum = $ 3x + 3 $ → $ x = \frac{\text{sum} - 3}{3} $
- For 4 consecutive numbers: Sum = $ 4x + 6 $ → $ x = \frac{\text{sum} - 6}{4} $

You can use these formulas for quick solutions!

Let me know if you'd like a version with even/odd numbers!