Study guide: 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.
Ideas to explain in your own words
- A doubling model multiplies the current amount by two each period. The initial amount is multiplied by a power whose exponent counts the number of completed periods. The model is supplied rather than a guarantee about a real population.
- A fixed percent decrease preserves the same fraction each period. The retained fraction multiplies the current amount, so absolute losses get smaller as the remaining amount shrinks. Losing twenty percent twice does not mean losing forty percent of the original.
- A geometric sequence multiplies each term by its common ratio. Starting from the first term, reaching term n requires n−1 repeated multiplications, which explains its explicit formula. Sequence indexing begins at the stated first term here.
- Equal-step outputs can be inspected for constant differences or constant nonzero ratios. A doubling table has a constant ratio while its differences grow, identifying exponential rather than linear structure. A few values can match many unrestricted formulas, so compare the supplied candidate rules rather than claiming uniqueness among all mathematical functions.
Useful vocabulary
- growth factor
- The multiplier applied at each equal period.
- geometric sequence
- A sequence with a constant ratio between consecutive terms.
- initial value
- A model’s output at its stated starting input.
Watch your reasoning
- The model is supplied rather than a guarantee about a real population. Include period zero as the initial amount; adding the multiplier instead of repeatedly multiplying creates a different pattern.
- Losing twenty percent twice does not mean losing forty percent of the original. Track the surviving multiplier at each period, keeping sufficient digits until the final requested rounding.
- Sequence indexing begins at the stated first term here. An exponent n would count one extra multiplication; write the first few terms to check the formula and avoid an off-by-one error.
A worked review example
Complete supplied guide examples: Count Repeated Multiplications in Growth — A fictional model doubles each period from 20. What is the amount after 3 full periods? Candidate answers: (A) 159 (B) 160 (C) 161 (D) 320. The model is A(t)=20·2^t. Three multiplicative steps give 20×2×2×2=160; adding 2 three times would not describe doubling. Retain a Fraction of the Current Amount — A fictional model loses 20% each period from 200. What remains after two periods? Candidate answers: (A) 127 (B) 128 (C) 129 (D) 256. Each period retains 80%, so multiply by 0.8 twice: 200×0.8²=128. The second loss is based on the smaller current amount. Distinguish a Term Number from a Period Count — A geometric sequence starts with a₁=2 and ratio 2. Find a_3. Candidate answers: (A) 16 (B) 7 (C) 8 (D) 9. From term 1 to term 3 there are 2 multiplications by 2. Use a_n=2·2^(n−1) to obtain 8. Compare Differences with Ratios — The outputs at inputs 0,1,2,3 are 2,4,8,16. Which rule describes them? Candidate answers: (A) y=2+2x (B) y=2x² (C) y=2·2^x (D) y=2·3^x. Consecutive outputs have constant ratio 2, while differences change. At x=0 the output is 2, so y=2·2^x matches every supplied pair.
Make a practice plan
- Study the four methods in the listed order; attempt each independent check before opening its explanation.
- Complete each lesson’s three changed-input questions and record the operation or reasoning step behind any correction.
- Take the four-question unit review, then solve the transfer task with written justifications.
- After correcting errors, attempt the separate unit exam and revisit the linked lesson for any unresolved method.