Sequence and Series: AP and GP

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

A sequence is an ordered list of numbers following a fixed rule, while a series is the sum of the terms of that sequence. An Arithmetic Progression (AP) has a constant common difference d, and a Geometric Progression (GP) has a constant common ratio r (r ≠ 0).

Formula	Expression	Use
AP nth term	Tₙ = a + (n − 1)d	Find any term
AP sum	Sₙ = n/2 [2a + (n − 1)d]	Add n terms
GP nth term	Tₙ = arⁿ⁻¹	Find any term
GP sum (r ≠ 1)	Sₙ = a(1 − rⁿ)/(1 − r)	Add n terms
GP sum to infinity	S∞ = a/(1 − r)	Only when \|r\| < 1

AM of a and b is (a + b)/2; GM is √(ab).
For NECO, expect 2-mark short answers and a 5–6 mark structured problem.

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Recognising the Progression

Test consecutive pairs: if T₂ − T₁ = T₃ − T₂, the sequence is AP with d = that constant. If T₂/T₁ = T₃/T₂ = r, the sequence is GP. NECO SSCE Paper 1 frequently begins with this recognition step before asking for the nth term or sum.

Deriving the Key Formulas

Starting from T₁ = a and applying the rule repeatedly, T₂ = a + d, T₃ = a + 2d, … so Tₙ = a + (n − 1)d. Summing the n terms in pairs from opposite ends of the list gives Sₙ = n/2 [2a + (n − 1)d], which also equals n/2 (first + last term). For a GP, multiplying a by r each step gives Tₙ = arⁿ⁻¹. Multiplying Sₙ by r and subtracting produces the closed form Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1; if r = 1 every term equals a so Sₙ = na.

Inserting Means Between Two Numbers

To insert m AMs between p and q, set a = p, find d from Tₘ₊₂ = q giving d = (q − p)/(m + 1), then list the terms. To insert m GMs between p and q, treat the block as a GP with first term p and (m + 2)th term q, giving r = (q/p)^(1/(m+1)).

Concept	Key point
AM ≥ GM	(a + b)/2 ≥ √(ab); equality only when a = b
Sum to infinity	Valid only when \|r\| < 1; otherwise S∞ diverges
Negative r	Terms alternate in sign but \|r\| < 1 still gives convergence
Common traps	Mixing up AM and GM; forgetting the −1 in n − 1

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Worked Example

Find the sum of the first 8 terms of the AP with first term 3 and common difference 4. Using Tₙ = a + (n − 1)d gives the 8th term as 3 + 7(4) = 31. Then S₈ = 8/2 (3 + 31) = 4 × 34 = 136. Cross-check with Sₙ = n/2 [2a + (n − 1)d] = 4 [6 + 7(4)] = 4 × 34 = 136. ✓

Edge Cases and Connections

A GP with any zero term forces all later terms to be 0, so S∞ and division-based sums break down — only the finite sum formula or Sₙ = na applies.
When r = −1/2 and a = 4, S∞ = 4/(1 + 1/2) = 8/3, even though individual terms alternate: 4, −2, 1, −0.5, …
AP/GP link closely with compound interest (r = 1 + i), depreciation (r = 1 − i), and population models, which is why NECO Paper 2 loves these applications.
The arithmetic–geometric mean inequality, AM ≥ GM, is a useful sanity check: if AM ≠ GM numerically, you have swapped the formulas.

Practice Prompts

The 5th and 12th terms of an AP are 21 and 49. Find the sum of the first 20 terms.
A bouncing ball rises to 80% of its previous height. If the initial drop is 10 m, find the total vertical distance travelled before the ball comes to rest (use S∞ with r = 0.8).

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Sequence and Series: AP and GP