Sequences and Series — AP, GP and Sum Formulas

What Is a Sequence?

A sequence is an ordered list of numbers that follow a specific pattern or rule. Each number in a sequence is called a term. The first term is usually denoted a (or a₁), and subsequent terms are a₂, a₃, a₄, and so on. The nth term is written as aₙ or T(n).

37111519...

The sequence above increases by 4 each time — this is an example of an Arithmetic Progression.

A series is what you get when you add the terms of a sequence together. If the sequence is 3, 7, 11, 15…, then the series is 3 + 7 + 11 + 15 + … and we are usually asked to find the sum of the first n terms.

Arithmetic Progression (AP)

Each term is obtained by adding a constant value (called the common difference, d) to the previous term.

Example: 2, 5, 8, 11, 14 ... (d = 3)

Example: 10, 7, 4, 1, −2 ... (d = −3)

Geometric Progression (GP)

Each term is obtained by multiplying by a constant value (called the common ratio, r) to the previous term.

Example: 2, 6, 18, 54 ... (r = 3)

Example: 100, 50, 25, 12.5 ... (r = 0.5)

Part 1 — Arithmetic Progressions (AP)

An Arithmetic Progression is a sequence where the difference between consecutive terms is constant. This constant difference is called d (the common difference). The first term is called a.

Identifying an AP

To check if a sequence is an AP, calculate the difference between consecutive terms. If the difference is always the same, it is an AP and that difference is d.

For the sequence 4, 9, 14, 19, 24: differences are 9−4=5, 14−9=5, 19−14=5. Since d=5 is constant, this is an AP with a=4, d=5.

AP Formulas

nth term: Tₙ = a + (n − 1)d

Sum of first n terms: Sₙ = n/2 × [2a + (n − 1)d]

Alternative sum formula: Sₙ = n/2 × (a + l)

where a = first term, d = common difference, n = number of terms, l = last term

Example 1 — Find the nth term of an AP

An AP has first term 5 and common difference 3. Find the 10th term.

Write down: a = 5, d = 3, n = 10

Apply formula: T₁₀ = 5 + (10 − 1) × 3

T₁₀ = 5 + 9 × 3 = 5 + 27 = 32

The 10th term is 32

Example 2 — Find the sum of an AP

Find the sum of the first 15 terms of the AP: 3, 7, 11, 15, ...

Identify: a = 3, d = 4, n = 15

Apply: S₁₅ = 15/2 × [2(3) + (15−1)(4)]

= 15/2 × [6 + 56] = 15/2 × 62 = 15 × 31 = 465

S₁₅ = 465

Example 3 — Find a and d from two terms

In an AP, the 4th term is 17 and the 9th term is 37. Find a and d.

T₄ = a + 3d = 17 ... (equation 1)

T₉ = a + 8d = 37 ... (equation 2)

Subtract equation 1 from equation 2: 5d = 20 → d = 4

Substitute into equation 1: a + 12 = 17 → a = 5

a = 5, d = 4. The AP is 5, 9, 13, 17, 21, 25, 29, 33, 37...

Part 2 — Geometric Progressions (GP)

A Geometric Progression is a sequence where each term is found by multiplying the previous term by a constant called the common ratio r. The first term is still called a.

Identifying a GP

To check if a sequence is a GP, divide each term by the previous term. If the ratio is always the same, it is a GP and that ratio is r.

For 2, 6, 18, 54, 162: ratios are 6/2=3, 18/6=3, 54/18=3. Since r=3 is constant, this is a GP with a=2, r=3.

GP Formulas

nth term: Tₙ = arⁿ⁻¹

Sum (r ≠ 1): Sₙ = a(rⁿ − 1) / (r − 1) when r > 1

Sum (r ≠ 1): Sₙ = a(1 − rⁿ) / (1 − r) when r < 1

Sum to infinity (|r| < 1): S∞ = a / (1 − r)

where a = first term, r = common ratio, n = number of terms

Example 4 — Find the nth term of a GP

A GP has first term 3 and common ratio 2. Find the 6th term.

Write down: a = 3, r = 2, n = 6

Apply: T₆ = 3 × 2⁶⁻¹ = 3 × 2⁵ = 3 × 32 = 96

The 6th term is 96

Check by listing: 3, 6, 12, 24, 48, 96 ✓

Example 5 — Sum of a GP

Find the sum of the first 5 terms of the GP: 2, 6, 18, 54, 162

Identify: a = 2, r = 3, n = 5

Since r > 1: S₅ = 2(3⁵ − 1) / (3 − 1) = 2(243 − 1) / 2 = 242

S₅ = 242

Check: 2+6+18+54+162 = 242 ✓

Example 6 — Sum to infinity

A GP has a = 8 and r = 1/2. Find the sum to infinity.

Since |r| = 1/2 < 1, the sum to infinity exists

S∞ = a / (1 − r) = 8 / (1 − 1/2) = 8 / (1/2) = 16

S∞ = 16

The sequence 8 + 4 + 2 + 1 + 0.5 + ... adds up to exactly 16

AP vs GP — How to Tell the Difference

If you subtract consecutive terms and get a constant → AP (common difference d)
If you divide consecutive terms and get a constant → GP (common ratio r)
If the sequence increases by a fixed amount each time → AP
If the sequence doubles, triples or halves each time → GP

KCSE Examination Tips

Memorise the four key formulas: AP nth term, AP sum, GP nth term, GP sum.
Always identify a, d (or r) and n first before applying any formula. Write them down explicitly.
For AP problems with two unknowns (find a and d), set up two simultaneous equations using the nth term formula and solve.
Sum to infinity only exists for GPs where the common ratio r satisfies −1 < r < 1. If |r| ≥ 1, the series diverges and has no finite sum.
Show your substitution step — write the formula, then substitute, then simplify. This earns method marks.

∑ Practice Sequences

Test yourself on AP terms, GP terms and series sums with our Sequences & Series game.

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