How to Solve: Arithmetic Progression (AP)

How to Solve: Arithmetic Progression (AP)

Introduction

"Imagine you’re saving pocket money—$5 this week, $8 next week, $11 the week after. How much will you have in 10 weeks? Arithmetic Progression (AP) is the secret formula that answers this—and it’s a guaranteed 3-5 marks on your exam!

What You Need To Know First

Before diving into AP, ensure you understand: 1. Basic algebra – Solving for unknowns (e.g., 2x + 3 = 7). 2. Sequences – What a sequence is (a list of numbers following a pattern). 3. Linear equations – Writing and solving equations like y = mx + c.

Key Vocabulary

Formulas To Know

1. nth Term Formula

Formula: [ T_n = a + (n - 1)d ]

Variables: - ( T_n ) = nth term - ( a ) = first term - ( d ) = common difference - ( n ) = term number

Memorise? ✅ MEMORISE THIS (Not always given on exam sheets!)

2. Sum of First n Terms (Sₙ)

Formula (Option 1): [ S_n = \frac{n}{2} [2a + (n - 1)d] ]

Formula (Option 2): [ S_n = \frac{n}{2} (a + T_n) ] (Use this if you already know the nth term!)

Variables: - ( S_n ) = sum of first n terms - ( a ) = first term - ( T_n ) = nth term - ( d ) = common difference - ( n ) = number of terms

Memorise? ✅ MEMORISE THIS (Sometimes given, but not always.)

Step-by-Step Method

Step 1: Identify the First Term (a) and Common Difference (d)

• First term (a): The first number in the sequence.
• Common difference (d): Subtract any term from the term after it. Example: In 4, 7, 10…, d = 7 - 4 = 3.

Step 2: Decide What You Need to Find

• nth term? → Use ( T_n = a + (n - 1)d ).
• Sum of terms? → Use ( S_n = \frac{n}{2} [2a + (n - 1)d] ) or ( S_n = \frac{n}{2} (a + T_n) ).
• Number of terms (n)? → Rearrange the nth term formula.

Step 3: Plug Values into the Correct Formula

• Write the formula.
• Substitute a, d, and n (or Tₙ).
• Solve for the unknown.

Step 4: Check Your Answer

• Does the answer make sense? Example: If d is positive, terms should increase. If negative, they should decrease.
• For sums, verify with a small n (e.g., sum of first 2 terms should match a + (a + d)).

Worked Example (Using Steps Above)

Problem: Find the 10th term and the sum of the first 10 terms of the AP: 2, 5, 8, 11…

Step 1: Identify a and d - First term (a) = 2 - Common difference (d) = 5 - 2 = 3

Step 2: Find the 10th term (T₁₀) - Use ( T_n = a + (n - 1)d ) - ( T_{10} = 2 + (10 - 1) \times 3 ) - ( T_{10} = 2 + 27 = 29 )

Step 3: Find the sum of first 10 terms (S₁₀) - Use ( S_n = \frac{n}{2} (a + T_n) ) - ( S_{10} = \frac{10}{2} (2 + 29) ) - ( S_{10} = 5 \times 31 = 155 )

Step 4: Check - 10th term should be 29 (2, 5, 8, 11, 14, 17, 20, 23, 26, 29) ✅ - Sum of first 2 terms = 2 + 5 = 7. Using formula: ( S_2 = \frac{2}{2} (2 + 5) = 7 ) ✅

Worked Examples

Example 1 – Basic: Find the nth Term

Problem: Find the 15th term of the AP: 7, 12, 17, 22…

Solution: 1. a = 7, d = 12 - 7 = 5 2. Use ( T_n = a + (n - 1)d ) 3. ( T_{15} = 7 + (15 - 1) \times 5 ) 4. ( T_{15} = 7 + 70 = 77 )

What we did and why: - We identified a and d first. - Used the nth term formula because we needed a specific term. - Substituted n = 15 to find the 15th term.

Example 2 – Medium: Find the Number of Terms

Problem: How many terms are in the AP: 3, 8, 13…, 98?

Solution: 1. a = 3, d = 8 - 3 = 5, Tₙ = 98 2. Use ( T_n = a + (n - 1)d ) 3. 98 = 3 + (n - 1) × 5 4. 98 - 3 = (n - 1) × 5 5. 95 = 5(n - 1) 6. 19 = n - 1 7. n = 20

What we did and why: - We knew the last term (Tₙ) and needed to find n. - Rearranged the nth term formula to solve for n. - Always check: 3 + (20 - 1) × 5 = 3 + 95 = 98 ✅

Example 3 – Exam Style: Sum of Terms with a Twist

Problem: The sum of the first 6 terms of an AP is 96. The first term is 5. Find the common difference.

Solution: 1. S₆ = 96, a = 5, n = 6 2. Use ( S_n = \frac{n}{2} [2a + (n - 1)d] ) 3. 96 = ( \frac{6}{2} [2 \times 5 + (6 - 1)d] ) 4. 96 = 3 [10 + 5d] 5. 32 = 10 + 5d 6. 22 = 5d 7. d = 4.4

What we did and why: - We used the sum formula because Sₙ was given. - Substituted known values and solved for d. - Always verify: If d = 4.4, the first 6 terms are 5, 9.4, 13.8, 18.2, 22.6, 27. Sum = 96 ✅

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, let’s lock this in—tonight, before your exam. Arithmetic Progression is just a sequence where you add the same number every time. That number is d, the common difference. The first term is a.

To find any term: Use ( T_n = a + (n - 1)d ). Plug in a, d, and n—done.

To find the sum of terms: Use ( S_n = \frac{n}{2} [2a + (n - 1)d] ). Or, if you know the last term, use ( S_n = \frac{n}{2} (a + T_n) ).

Watch out for: - Negative d (sequence goes down). - Word problems—extract a and d first. - Non-integer answers for n—if n isn’t a whole number, the term doesn’t exist.

Last tip: Always write the formula first, then substitute. Double-check d by subtracting two terms. You’ve got this—go ace that exam!