How to Solve: Sequence and Series (AP, GP, HP)

How to Solve: Sequence and Series (AP, GP, HP)

(For SSC, Bank, Railway Exams – 1200+ Words)

Introduction

"Mastering Arithmetic, Geometric, and Harmonic Progressions can add 5–10 marks to your SSC/Bank/Railway exam—enough to push you from ‘just passing’ to ‘top rank.’ These questions appear in every paper, and if you follow this exact method, you’ll solve them in under 60 seconds."

What You Need To Know First

1. Basic algebra – Solving linear equations (e.g., 2x + 3 = 7).
2. Mean concepts – Arithmetic mean (AM), Geometric mean (GM), Harmonic mean (HM).
3. Summation notation – What Σ (sigma) means (e.g., Σaₙ = sum of terms).

Key Vocabulary

Formulas To Know

1. Arithmetic Progression (AP)

2. Geometric Progression (GP)

3. Harmonic Progression (HP)

Step-by-Step Method

Step 1: Identify the Type of Sequence

• AP Check: Difference between terms is constant (d). Example: 5, 9, 13, 17 → d = 4.
• GP Check: Ratio between terms is constant (r). Example: 3, 6, 12, 24 → r = 2.
• HP Check: Reciprocals form an AP. Example: 1, 1/2, 1/3, 1/4 → Reciprocals: 1, 2, 3, 4 (AP).

Step 2: Write Down Known Values

• For AP: a (first term), d (common difference), n (term number).
• For GP: a (first term), r (common ratio), n (term number).
• For HP: Convert to AP first (take reciprocals).

Step 3: Choose the Right Formula

• Need the nth term? → Use aₙ formula.
• Need the sum of terms? → Use Sₙ formula.
• Need mean? → Use AM/GM/HM formula.

Step 4: Plug in Values and Solve

• Substitute known values into the formula.
• Solve for the unknown (e.g., n, a, d, r).

Step 5: Verify the Answer

• Check if the answer makes sense (e.g., n must be a positive integer).
• For GP, ensure r is consistent.

Worked Examples

Example 1 – Basic AP

Question: Find the 10th term of the AP: 2, 5, 8, 11, …

Solution: 1. Identify type: AP (difference = +3). 2. Known values: a = 2, d = 3, n = 10. 3. Formula: aₙ = a + (n-1)d. 4. Plug in: a₁₀ = 2 + (10-1)×3 = 2 + 27 = 29. 5. Verify: 10th term should be 29 (count terms to confirm).

What we did and why: - We used the nth term formula for AP because we needed a specific term. - n-1 is used because the first term is already a.

Example 2 – Medium GP

Question: Find the sum of the first 6 terms of the GP: 3, 6, 12, 24, …

Solution: 1. Identify type: GP (ratio = ×2). 2. Known values: a = 3, r = 2, n = 6. 3. Formula: Sₙ = a(1 - rⁿ)/(1 - r) (since r > 1). 4. Plug in: S₆ = 3(1 - 2⁶)/(1 - 2) = 3(1 - 64)/(-1) = 3(-63)/(-1) = 189. 5. Verify: Sum of 3 + 6 + 12 + 24 + 48 + 96 = 189.

What we did and why: - We used the sum formula for GP because r > 1. - The denominator (1 - r) becomes negative, so signs cancel out.

Example 3 – Exam-Style HP

Question: If 1/a, 1/b, 1/c are in AP, and a, b, c are in HP, find b if a = 2 and c = 6.

Solution: 1. Identify type: HP → reciprocals form AP. 2. Convert to AP: 1/a, 1/b, 1/c are in AP. 3. AP property: 2/b = 1/a + 1/c (middle term = average of neighbors). 4. Plug in: 2/b = 1/2 + 1/6 = (3 + 1)/6 = 4/6 = 2/3. 5. Solve for b: 2/b = 2/3 → b = 3. 6. Verify: Check if 1/2, 1/3, 1/6 are in AP (difference = -1/6).

What we did and why: - We used the AP mean property because HP’s reciprocals form an AP. - The key was recognizing that 2/b is the average of 1/a and 1/c.

Common Mistakes

Exam Traps

1-Minute Recap

"Listen up—this is your last-minute cheat sheet for AP, GP, and HP. First, identify the sequence: AP if terms add a fixed number, GP if they multiply, HP if reciprocals form an AP. For AP, memorize the nth term (a + (n-1)d) and sum (n/2 [2a + (n-1)d]). For GP, nth term is (a × r^(n-1)), sum is (a(1 - rⁿ)/(1 - r)). HP? Flip it to AP first. Always write down known values before plugging into formulas. Watch out for traps: negative ratios, disguised sequences, and non-integer terms. If you see fractions, think HP. Now go crush those 5–10 marks!