Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Browse grades and teaching videos01 · Read and understand
What you will learn
- Identify the first term and common difference.
- Use a position formula with n − 1 increases.
- Sum a finite arithmetic sequence by pairing terms.
Before you start
Substitute into linear expressions and distinguish a single term from a running total.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
A fictional auditorium layout has 12 seats in its first row and 3 additional seats in each successive row. How many seats are in row 10, and in the first 10 rows combined?
Why this math matters
A repeated addition can be expressed as a rule for any position. Careful indexing matters: the first term has experienced no increases yet. Summing the terms answers a different question from finding just the last row.
Set up the model
A useful answer starts with clear assumptions:
- Rows are numbered starting at one.
- The stated arithmetic pattern continues through all ten rows without aisles removing seats.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Predict seats in a growing row pattern
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A fictional auditorium layout has 12 seats in its first row and 3 additional seats in each successive row. How many seats are in row 10, and in the first 10 rows combined?
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Describe row n
aₙ = 12 + 3(n − 1)
Row two adds three once; row three adds three twice. Row n therefore includes n minus one increases.
Find row ten
a₁₀ = 12 + 3(9) = 39 seats
There are nine gaps between rows one and ten. Using ten increases would incorrectly describe row eleven.
Add all ten rows
total = 10(12 + 39)/2 = 255 seats
Pair the first and last rows, the second and ninth, and so on. Each of the five pairs contains fifty-one seats.
The result
Row 10 contains 39 seats; the first 10 rows contain 255 seats altogether.
The average of the first and last terms is 25.5. Multiplying by ten gives the total even though no actual row has a fractional seat count. An average need not be an observed term.
Common mistakes to catch
- Using n instead of n − 1 shifts the sequence by one position.
- Multiplying the final row's count by ten assumes every row is as large as the last.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
A row pattern begins with 8 seats and adds 2 per row. Find row 15.
Show a hint
Fifteen rows create fourteen increases.
Reveal answer and explanation
36 seats
8 + 2(15 − 1) = 8 + 28 = 36.
Practice 2
Row 7 contains 30 seats, and each row adds 4. How many seats are in row 1?
Show a hint
Move back through six increases.
Reveal answer and explanation
6 seats
30 − 6(4) = 6. Forward checking gives 6 + 24 = 30 in row seven.
Take the idea with you
For any sequence formula, test n = 1 before using a distant term. That quickly catches many indexing errors.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Understand why more groups mean fewer supplies per group
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