Mathematics · 16 / 23
Sequences: add or multiply each step
High school · intermediate · Worked example and 4 exercises with solutions, at your pace.
The principle, simply
A sequence gives values indexed by integers. An arithmetic sequence adds a constant difference; a geometric sequence multiplies by a constant ratio. A recurrence uses the previous term, whereas an explicit formula calculates a term directly. State the initial index.
- 01Initial term
- 02Step rule
- 03Required index
- 04Formula or recurrence
WORKED EXAMPLE
From principle to reasoning
Starting at u₀ = 100 and increasing 5% each step gives uₙ₊₁ = 1.05uₙ and uₙ = 100 × 1.05ⁿ. After two steps, u₂ = 110.25.
Solve a problem in this subject
- Define the objects
State unknowns, domain and data. Real square roots need nonnegative arguments; denominators must be nonzero.
- Choose a transformation
Name the rule: expansion, equivalent equations, factoring, differentiation. Check its conditions. Squaring an equation requires testing candidates in the original.
- Show the reasoning
Each line should follow from the previous one. Keep exact values where possible and distinguish equality, approximation and implication.
- Test and conclude
Substitute into the original expression. Check graph points and domain boundaries. State a solution set or the required unit.
YOUR TURN
01 · Try before checking
Starting at u₀ = 7 and adding 3, find u₄.
One hint
Four steps connect index 0 to index 4.
Reveal and understand the solution
uₙ = 7 + 3n, so u₄ = 19. Listing gives 7, 10, 13, 16, 19.
Compare method, conditions and answer; identify the exact part to revisit.
IMPROVE THROUGH PRACTICE
Three additional challenges with solutions
Start with the first challenge. Write your method before opening a hint; use the solution to find the first step that needs work.
u₀ = 7 and uₙ₊₁ = uₙ + 3. Find u₄.
A hint to get started
From index 0 to 4 there are four additions.
Step-by-step solution
- This is arithmetic with common difference 3.
- u₄ = 7 + 4 × 3.
- u₄ = 19; the list u₀ through u₄ contains five terms.
Which step did you miss? Retry without help, then explain why the method works.
A stock of 200 falls by 10% per period. Find it after three periods.
A hint to get started
A 10% decrease multiplies by 0.9.
Step-by-step solution
- uₙ = 200 × 0.9ⁿ.
- u₃ = 200 × 0.729.
- The continuous model gives 145.8 units.
Which step did you miss? Retry without help, then explain why the method works.
For uₙ = 2 + 1/(n + 1), n ≥ 0, find its variation and limit.
A hint to get started
Inspect the positive term added to 2.
Step-by-step solution
- 1/(n + 1) decreases and stays positive.
- uₙ decreases while staying above 2.
- The added term tends to zero, so the limit is 2. A sequence need not reach its limit.
Which step did you miss? Retry without help, then explain why the method works.
The trap to avoid
Four terms and four changes are different; check the first index.
Without notes, can you…
- Explain the principle in your own words?
- Rebuild the diagram and its connections?
- Solve the exercise and justify the method?
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