How do I quickly identify whether a sequence is AP or GP?

Compute the differences between consecutive terms. If constant, it is an AP. If not, compute the ratios — if constant, it is a GP. AP test: Tₙ − Tₙ₋₁ = d. GP test: Tₙ / Tₙ₋₁ = r.

When does the sum to infinity of a GP exist?

Only when |r| < 1 (equivalently, −1 < r < 1). If |r| ≥ 1 the terms do not approach zero and the series diverges. The sum S∞ = a/(1 − r) applies exclusively when the convergence condition is met.

How do I find n when I know the sum of n terms?

Substitute Sₙ = given value into the AP or GP sum formula and solve for n. For AP this is usually a linear equation; for GP you may need logarithms to isolate n. Remember: n must be a positive integer — always round up from decimal results (e.g., n > 12.3 → n = 13).

Arithmetic & Geometric Sequences

Q: When does the sum to infinity of a GP exist?

Only when |r| < 1 (equivalently, −1 < r < 1). If |r| ≥ 1 the terms do not approach zero and the series diverges. The sum S∞ = a/(1 − r) applies exclusively when the convergence condition is met.

Q: How do I find n when I know the sum of n terms?

Substitute Sₙ = given value into the AP or GP sum formula and solve for n. For AP this is usually a linear equation; for GP you may need logarithms to isolate n. Remember: n must be a positive integer — always round up from decimal results (e.g., n > 12.3 → n = 13).

Master AP and GP formulas, find the general term (Tₙ) and sum of terms (Sₙ), and conquer the convergence constraints of the sum to infinity (S∞).

1. The Core Concept

The Pattern of Numbers: A sequence is an ordered list of numbers. In an Arithmetic Progression (AP), you add a constant difference (d) to get the next term. In a Geometric Progression (GP), you multiply by a constant ratio (r) to get the next term.

2. DSE Arithmetic & Geometric Sequences Cheat Sheet — 3 Key Topics

Scan this compact reference table to review every essential concept, step, and shortcut in under 10 seconds.

Topic	Key Concept & Steps	Exam Shortcuts & Tips
1. Arithmetic (AP)	General Term: `Tₙ = a + (n − 1)d` (where `a` = 1st term, `d` = common diff.)	Sum to n Terms: • General: `Sₙ = (n/2)[2a + (n − 1)d]` • If last term `l` is known: `Sₙ = (n/2)(a + l)` AP Test: `Tₙ − Tₙ₋₁ = d` (The difference between any two consecutive terms is constant)
2. Geometric (GP)	General Term: `Tₙ = arⁿ⁻¹` (where `a` = 1st term, `r` = common ratio)	Sum to n Terms: • If `r` < 1: `Sₙ = a(1 − rⁿ) / (1 − r)` • If `r` > 1: `Sₙ = a(rⁿ − 1) / (r − 1)` GP Test: `Tₙ / Tₙ₋₁ = r` (The ratio of any two consecutive terms is constant)
3. Sum to Infinity	Formula: `S∞ = a / (1 − r)`	Convergence Condition: The sum to infinity exists only if: `−1 < r < 1` ⟺ `\|r\| < 1` If `r` ≥ 1 or `r` ≤ −1, the series diverges and S∞ does not exist.

💡 Exam Traps & Secrets

⚠️ The "Number of Terms (n)" Integer ConstraintBecause n represents the physical count of terms, n must always be a positive integer (n ∈ ℤ⁺). Inequality Trap: If a question asks for "the first n terms whose sum exceeds 500," and your inequality yields n > 12.3, the smallest valid integer is n = 13 (not 12!).

⚠️ The Log inequality Sign Flip (GP Trap)When solving for n in a GP inequality where r is a decimal (e.g., 0.6ⁿ⁻¹ < 0.01), taking the logarithm on both sides flips the inequality sign because log(0.6) is a negative number! (n − 1) log(0.6) < log(0.01) ⟹ n − 1 > log(0.01) / log(0.6)

⚠️ The Sum to Infinity Existence Trap (Paper 2)DSE Multiple-Choice questions love to ask: "For what range of x does the sum to infinity of the GP 4, 4(x − 2), 4(x − 2)², … exist?" Shortcut: Identify the common ratio r = x − 2. Apply the convergence condition |r| < 1 ⟹ |x − 2| < 1. Solve the absolute value sandwich: −1 < x − 2 < 1 ⟹ 1 < x < 3.

✏️ Self-Test: Quick Interactive Practice

How to study: Click any question below to instantly load it on the coordinate grid and check your steps.

(Setting up and solving simultaneous linear equations)The 4th term of an AP is 14 and the 9th term is 34. Find the first term (a) and the common difference (d).

(Solving exponential equations derived from Sₙ)The sum of the first n terms of a GP is 255. If the first term is 1 and the common ratio is 2, find n.

(Convergence validation & S∞ calculation)A GP has a first term of 80 and a common ratio of 0.6. Find its sum to infinity.

Frequently Asked Questions

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