NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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A pattern of dots grows such that the number of dots in each stage follows a linear rule. The first stage has 4 dots, the second has 9 dots, and the third has 14 dots.

The rule for the number of dots, y, in stage x is y = 5x - 1.

What is the exact number of dots in stage 27?

Concept illustrations

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A straight-line relationship. Illustration: y = x + 1. Each unit right adds one unit up; the y-intercept is 1.-2-2-1-111223344xy
Illustration: y = x + 1. Each unit right adds one unit up; the y-intercept is 1.
Keep both sides balanced. The level beam represents equality, not measured weights. Subtract 6 from both sides of x + 6 = 14 to leave x = 8.x + 614−6−6x = 8
The level beam represents equality, not measured weights. Subtract 6 from both sides of x + 6 = 14 to leave x = 8.

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Formula

Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. The pattern of dots follows a linear rule given by the formula y = 5x - 1, where y is the number of dots and x is the stage number.
  2. We are asked to find the number of dots in stage 27, so substitute x = 27 into the formula.
  3. Substitute 27 for x: y = 5 * 27 - 1.
  4. Multiply 5 by 27: 5 * 27 = 135.
  5. Subtract 1 from 135: 135 - 1 = 134.
  6. The number of dots in stage 27 is 134.
  7. Verify the rule using known values: for stage 1, y = 5*1 - 1 = 4 (matches given).
  8. Check stage 2: y = 5*2 - 1 = 10 - 1 = 9 (matches given).
  9. Check stage 3: y = 5*3 - 1 = 15 - 1 = 14 (matches given).
  10. The rule is consistent with the given pattern.
  11. Therefore, the number of dots in stage 27 is exactly 134.
  12. Final answer confirmed by substitution and verification.

Reference answer: 134

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