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Grade 9 · unit review

Review: Algebra1: Exponential Patterns and Geometric Sequences

An exponential pattern changes by a constant multiplicative ratio over equal input steps. This differs from a linear pattern’s constant addition and can model supplied growth, decay or ordered geometric sequence terms.

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Bring the unit skills together

Use these 4 new cases to check your reasoning. Choices are temporary and are not saved to an account, sent to staff or recorded as school grades. Download your current review choices before leaving or reloading.

Unit review questions
1. A fictional model doubles each period from 70. What is the amount after 3 full periods?
Need a hint?
  • A doubling model multiplies the current amount by two each period.
  • The model is supplied rather than a guarantee about a real population.
2. A fictional model loses 20% each period from 700. What remains after two periods?
Need a hint?
  • A fixed percent decrease preserves the same fraction each period.
  • Losing twenty percent twice does not mean losing forty percent of the original.
3. A geometric sequence starts with a₁=7 and ratio 2. Find a_8.
Need a hint?
  • A geometric sequence multiplies each term by its common ratio.
  • Sequence indexing begins at the stated first term here.
4. The outputs at inputs 0,1,2,3 are 7,14,28,56. Which rule describes them?
Need a hint?
  • Equal-step outputs can be inspected for constant differences or constant nonzero ratios.
  • A few values can match many unrestricted formulas, so compare the supplied candidate rules rather than claiming uniqueness among all mathematical functions.

The download contains your current choices and any checked feedback. Keep the separate transfer-task work in your own document; downloading its instructions does not mark it complete.

Use the skills in a new task

Create a four-part mathematical solution sheet for these supplied cases. 1. A fictional model doubles each period from 90. What is the amount after 3 full periods? Candidate answers: (A) 1440 (B) 719 (C) 720 (D) 721. 2. A fictional model loses 20% each period from 900. What remains after two periods? Candidate answers: (A) 1152 (B) 575 (C) 576 (D) 577. 3. A geometric sequence starts with a₁=9 and ratio 2. Find a_10. Candidate answers: (A) 4607 (B) 4608 (C) 4609 (D) 9216. 4. The outputs at inputs 0,1,2,3 are 9,18,36,72. Which rule describes them? Candidate answers: (A) y=9+2x (B) y=9x² (C) y=9·2^x (D) y=9·3^x.

Work on paper or in your own document. Explain the choices you made.

  • State each requested quantity or relationship and its input restrictions.
  • Show intermediate calculations or the applicable theorem with matching correspondences.
  • Give all four results, including exact values or requested rounding and units.
  • Check a result by substitution, reversal, dimensions, a representation or a counterexample as appropriate.
Compare with a model response

1. 720. The model is A(t)=90·2^t. Three multiplicative steps give 90×2×2×2=720; adding 2 three times would not describe doubling. 2. 576. Each period retains 80%, so multiply by 0.8 twice: 900×0.8²=576. The second loss is based on the smaller current amount. 3. 4608. From term 1 to term 10 there are 9 multiplications by 2. Use a_n=9·2^(n−1) to obtain 4608. 4. y=9·2^x. Consecutive outputs have constant ratio 2, while differences change. At x=0 the output is 9, so y=9·2^x matches every supplied pair.

A different response can also meet the criteria. Compare the evidence and reasoning, rather than copying the wording.