Mathematics Grade 10 Arithmetic Progressions nth Term and Sum

Grade 10 Mathematics: Arithmetic Progressions — nth Term and Sum

1. The Driving Question

You’re saving money for a new phone. Week 1, you save $10. Week 2, you save $15. Week 3, $20. If you keep adding $5 every week, how much will you have saved by Week 12 — and how can you predict it without adding up 12 numbers one by one? Why does this pattern work, and where else in life do these kinds of "step-by-step" increases show up?

2. The Core Idea — Built, Not Listed

Imagine you’re climbing a staircase where each step is exactly 6 inches taller than the one before it. The first step is 6 inches high. The second is 12 inches. The third is 18 inches. If you want to know how high the 20th step is, you don’t need to build 20 steps — you can predict it. That’s what an arithmetic progression (AP) does: it’s a sequence where each term increases (or decreases) by the same fixed amount, called the common difference.

In your savings example, the first term (a₁) is $10, and the common difference (d) is $5. The sequence is 10, 15, 20, 25… and so on. The nth term of an AP tells you the value at any position — like knowing how much you’ll save in Week n without counting every week. The sum of the first n terms tells you the total after n steps — like knowing your total savings after 12 weeks without adding 10 + 15 + 20 + … + 65.

This isn’t just about money or stairs. It’s about predictable change — the kind you see in monthly gym membership fees that increase by $2 every year, or the way a car’s odometer increases by the same distance every hour on a road trip.

Key Vocabulary:
- Arithmetic progression (AP): A sequence where each term after the first is found by adding a constant (the common difference) to the previous term.
Example: The sequence of house numbers on your street — 102, 104, 106, 108… — is an AP with d = 2.
Note (Grades 11–12+): In calculus, APs are the discrete version of linear functions; their sums relate to Riemann sums and integration.

• Common difference (d): The fixed amount added (or subtracted) to get the next term in an AP.
  Example: In the sequence of years when the Summer Olympics are held (2020, 2024, 2028…), d = 4.
  Note: If d is negative, the sequence decreases (e.g., a car slowing down by 5 mph every second).

• nth term (aₙ): The value of the term at position n in the sequence.
  Example: In the AP 3, 7, 11, 15…, the 5th term (a₅) is 19.
  Formula: aₙ = a₁ + (n – 1)d

• Sum of the first n terms (Sₙ): The total of all terms from a₁ to aₙ.
  Example: The sum of the first 4 terms of the AP 2, 5, 8, 11 is 2 + 5 + 8 + 11 = 26.
  Formula: Sₙ = n/2 (2a₁ + (n – 1)d) or Sₙ = n/2 (a₁ + aₙ)

3. Assessment Translation

How This Appears on Tests:
- State Standardized Tests (e.g., PARCC, SBAC): Multiple-choice questions asking for the nth term or sum, often with word problems (e.g., "A theater has 20 rows of seats. The first row has 15 seats, and each row after has 2 more seats than the previous. How many seats are in the 10th row?"). Short-answer questions may ask you to explain how you found the common difference or why the formula works.
- SAT/ACT: Rarely tests APs directly, but the logic appears in problems about linear patterns (e.g., "A sequence starts at 5 and increases by 3 each time. What is the 50th term?"). The SAT may ask you to derive the formula for the nth term from a given sequence.
- Classroom Assessments: Exit tickets or quizzes often ask you to: - Find the 10th term of an AP given a₁ and d.
- Calculate the sum of the first 15 terms.
- Identify the common difference from a sequence.
- Solve a word problem (e.g., "A runner increases her distance by 0.5 miles each week. If she runs 3 miles in Week 1, how far will she run in Week 8?").

What a Proficient Response Looks Like:
- Multiple Choice: If the question asks for the 7th term of the AP 4, 9, 14…, a proficient student selects 34 (not 39, a common distractor from miscounting n).
- Short Answer: For the prompt "Find the sum of the first 20 terms of the AP where a₁ = 6 and d = 3," a proficient response shows:

a₂₀ = 6 + (20 – 1)(3) = 6 + 57 = 63 S₂₀ = 20/2 (6 + 63) = 10 × 69 = 690 Answer: 690 A developing response might calculate a₂₀ correctly but forget to divide by 2 in the sum formula.

Model Student Response (Word Problem):
Prompt: A concert venue has 30 rows of seats. The first row has 20 seats, and each subsequent row has 4 more seats than the row before it. How many seats are in the 15th row? How many seats are there in total in the first 15 rows?

Proficient Response:

The number of seats forms an AP where a₁ = 20 and d = 4.
For the 15th row: a₁₅ = 20 + (15 – 1)(4) = 20 + 56 = 76 seats For the total seats in 15 rows: First, find a₁₅ (already done: 76).
Then, S₁₅ = 15/2 (20 + 76) = 7.5 × 96 = 720 seats Answer: 76 seats in the 15th row; 720 seats total in the first 15 rows.

4. Mistake Taxonomy

Mistake 1: Miscounting n in the nth Term Formula
- Question: Find the 8th term of the AP 5, 11, 17… - Common Wrong Response: a₈ = 5 + (8)(6) = 53 - Why It Loses Credit: The formula is aₙ = a₁ + (n – 1)d, not aₙ = a₁ + nd. The student forgot to subtract 1 from n.
- Correct Approach:

d = 11 – 5 = 6 a₈ = 5 + (8 – 1)(6) = 5 + 42 = 47

Mistake 2: Using the Wrong Sum Formula
- Question: Find the sum of the first 10 terms of the AP where a₁ = 2 and d = 5.
- Common Wrong Response: S₁₀ = 10/2 (2 + 5) = 35 - Why It Loses Credit: The student used Sₙ = n/2 (a₁ + d) instead of Sₙ = n/2 (a₁ + aₙ). They forgot to calculate a₁₀ first.
- Correct Approach:

a₁₀ = 2 + (10 – 1)(5) = 2 + 45 = 47 S₁₀ = 10/2 (2 + 47) = 5 × 49 = 245

Mistake 3: Misinterpreting Word Problems (Off-by-One Errors)
- Question: A staircase has 12 steps. The first step is 10 cm high, and each step after is 2 cm higher. What is the height of the 12th step? - Common Wrong Response: a₁₂ = 10 + (12)(2) = 34 cm - Why It Loses Credit: The student treated the first step as a₀ instead of a₁. The 12th step is a₁₂, not a₁₃.
- Correct Approach:

a₁₂ = 10 + (12 – 1)(2) = 10 + 22 = 32 cm

5. Connection Layer

1. Within Math: Arithmetic progressions → Linear functions
   Why: The nth term formula (aₙ = a₁ + (n – 1)d) is a linear equation in disguise. If you graph the terms of an AP (e.g., (1, 5), (2, 8), (3, 11)…), you get a straight line with slope d. Understanding APs helps you see why linear functions have constant rates of change.

2. Across Subjects: Arithmetic progressions → Physics (uniform acceleration)
   Why: When a car accelerates at a constant rate (e.g., 2 m/s²), its velocity at each second forms an AP (e.g., 0, 2, 4, 6… m/s). The sum of the first n terms gives the total distance traveled — just like the sum formula in math.

3. Outside School: Arithmetic progressions → Subscription pricing models
   Why: Many apps (e.g., Spotify, Netflix) offer "tiered" pricing where each plan increases by a fixed amount (e.g., $5/month). The total cost over n months is the sum of an AP. If you understand APs, you can predict how much you’ll spend in a year without adding up 12 numbers.

6. The Stretch Question

If the sum of the first n terms of an AP is given by Sₙ = 3n² + 5n, what is the 10th term of the sequence?

Pointer Toward the Answer: The sum formula Sₙ is a quadratic in n, which makes sense because the sum of an AP grows quadratically (like n²). To find the 10th term (a₁₀), recall that aₙ = Sₙ – Sₙ₋₁. So:

a₁₀ = S₁₀ – S₉ S₁₀ = 3(10)² + 5(10) = 300 + 50 = 350 S₉ = 3(9)² + 5(9) = 243 + 45 = 288 a₁₀ = 350 – 288 = 62 But why does this work? Because Sₙ is the sum up to n, and Sₙ₋₁ is the sum up to n – 1, so their difference is the nth term. This trick is useful when you’re given the sum formula but not the sequence itself.