Sequence and Series: formula sheet

Key formulas

General term

tₙ = rule in n (e.g. tₙ = 3n + 2)

Put n = 1, 2, 3, ... to generate terms. n is a positive integer.

Series as a sum

Sₙ = t₁ + t₂ + ... + tₙ

Sₙ is the sum of the first n terms.

Term from sum

tₙ = Sₙ − Sₙ₋₁ (for n ≥ 2); t₁ = S₁

Use when the sum formula is given and you need a particular term.

Sum from terms

Sₙ = Sₙ₋₁ + tₙ

Use when you know the sum up to n − 1 and the nth term.

Common difference

d = T2 − T1 = Tn − T(n−1)

Must be the same for every consecutive pair.

nth term

Tn = a + (n − 1)d

a is the first term. Use it to find any term or the number of terms.

Sum of n terms (using d)

Sn = n/2 × [2a + (n − 1)d]

Use when a, d and n are known.

Sum of n terms (using last term)

Sn = n/2 × (a + l)

l is the last term, l = Tn. Use when the last term is known.

Term from sum

Tn = Sn − S(n−1)

Valid for n ≥ 2. T1 = S1.

Number of terms

n = (l − a) ÷ d + 1

Rearranged nth term formula. n must be a positive whole number.

Three terms in AP

If a, b, c are in AP, then 2b = a + c

The middle term is the average of its neighbours.

Selecting terms

3 terms: a − d, a, a + d. 4 terms: a − 3d, a − d, a + d, a + 3d. 5 terms: a − 2d, a − d, a, a + d, a + 2d

Symmetric choice makes the sum easy. For 4 terms the common difference is 2d.

Sum of first n natural numbers

1 + 2 + … + n = n(n + 1) ÷ 2

AP with a = 1, d = 1.

AM of two numbers

A = (a + b) ÷ 2

a, A, b are in AP. Works for any two real numbers.

Common difference after inserting n means

d = (b − a) ÷ (n + 1)

There are n + 1 equal gaps, not n.

The k-th inserted mean

Aₖ = a + k·d, for k = 1, 2, ..., n

A₁ = a + d is the first mean and Aₙ = b − d is the last.

Sum of the n inserted means

A₁ + A₂ + ... + Aₙ = n(a + b) ÷ 2

Equals n times the AM of a and b. Does not include a and b.

Three terms in AP

2B = A + C

Use this to test whether a middle term is the AM of its neighbours.

Common ratio

r = T₂ ÷ T₁ = T₃ ÷ T₂

Must be the same for every consecutive pair. Terms are non-zero.

nth term

Tₙ = a·r^(n−1)

The power is n − 1, not n.

Sum of n terms (r > 1)

Sₙ = a(rⁿ − 1) ÷ (r − 1)

Usually used when r is greater than 1, because the denominator is then positive. Valid for any r ≠ 1.

Sum of n terms (r < 1)

Sₙ = a(1 − rⁿ) ÷ (1 − r)

Usually used when r is less than 1 (r ≠ 1), because the denominator is then positive. It gives the same result as the other form.

Sum when r = 1

Sₙ = n·a

All terms are equal, so the formulas above do not apply.

Three terms in GP

b² = ac

b is the middle term. For a,b,c all non-zero in GP. Useful for finding a missing term.

Three terms in GP (assumed form)

a/r, a, ar

Handy when the product of three terms is given: the product is a³.

nth term from the end of a finite GP

l ÷ r^(n−1)

l is the last term. Equivalent to a GP with first term l and ratio 1/r.

Definition of HP

a, b, c are in HP ⇔ 1/a, 1/b, 1/c are in AP

All terms must be non-zero.

nth term of HP

Tn = 1 ÷ [1/a + (n − 1)d], where a is the first term of the HP and d = 1/T2 − 1/T1

d is the common difference of the reciprocal AP, not of the HP.

HM of two numbers

H = 2ab ÷ (a + b)

Also H = 2 ÷ (1/a + 1/b).

HM of n numbers

H = n ÷ (1/a1 + 1/a2 + ... + 1/an)

Numbers must be positive for the inequality to apply.

AM, GM, HM of two numbers

AM = (a + b)/2; GM = √(ab); HM = 2ab/(a + b)

Use for positive a and b.

Relation among means

GM² = AM × HM

Holds for two positive numbers. Not for three or more in general.

Inequality

AM ≥ GM ≥ HM

For positive numbers; equal only if all numbers are equal.

Sum of first n natural numbers

Σk = 1 + 2 + 3 + … + n = n(n + 1) ÷ 2

Works for k from 1 to n, where n is a positive integer.

Sum of squares of first n natural numbers

Σk² = 1² + 2² + … + n² = n(n + 1)(2n + 1) ÷ 6

Do not forget the division by 6. The result is always a whole number.

Sum of cubes of first n natural numbers

Σk³ = 1³ + 2³ + … + n³ = [n(n + 1) ÷ 2]²

Equal to (Σk)². Square the whole bracket, not just n.

Sum of a constant

Σc (k = 1 to n) = n × c

You are adding the constant c, n times.

Linearity of sigma

Σ(a·uₖ + b·vₖ) = a·Σuₖ + b·Σvₖ

Constants come out. Sums of terms split. This does not work for products or powers of sums.

Sum from m to n

Σ (k = m to n) f(k) = Σ (k = 1 to n) f(k) − Σ (k = 1 to m − 1) f(k)

Use when the lower limit is not 1.

Sum of first n odd numbers

1 + 3 + 5 + … + (2n − 1) = n²

Follows from Σ(2k − 1) = 2·n(n+1)/2 − n.

Sum of first n even numbers

2 + 4 + 6 + … + 2n = n(n + 1)

Follows from 2·Σk.

Sum of k(k + 1)

1·2 + 2·3 + … + n(n + 1) = n(n + 1)(n + 2) ÷ 3

Derived by splitting into Σk² + Σk. Useful as a check in options.

nth term of an AP

Tₙ = a + (n − 1)d

a is the first term, d is the common difference, n is the number of terms.

Sum of n terms of an AP

Sₙ = (n ÷ 2) × [2a + (n − 1)d]

Use it for savings plans where the deposit rises by a fixed amount.

nth term of a GP

Tₙ = a × r^(n − 1)

r is the common ratio. The first term is a, not a × r.

Sum of n terms of a GP

Sₙ = a(rⁿ − 1) ÷ (r − 1) for r > 1; Sₙ = a(1 − rⁿ) ÷ (1 − r) for r < 1

Both forms are equal. Pick the one with a positive denominator. Not valid for r = 1.

Simple interest

I = P × i × n; A = P(1 + n × i)

Yearly amounts form an AP with common difference P × i.

Compound interest

A = P(1 + i)ⁿ; CI = A − P

i is the rate per period as a decimal. Yearly amounts form a GP with ratio (1 + i).

Growth at a fixed percentage

Pₙ = P₀(1 + g)ⁿ

Use for population, sales or production growing g per period.

Reducing balance depreciation

Vₙ = C(1 − d)ⁿ

C is the cost, d is the yearly rate as a decimal. Values form a GP with ratio (1 − d).

Straight-line depreciation

Vₙ = C − n × D, where D = (C − scrap value) ÷ life

Values form an AP with common difference −D.

Sum of an infinite GP

S∞ = a ÷ (1 − r), valid only when |r| < 1

Used when a repeating, shrinking amount continues indefinitely.

Quick revision

• AP: nth term Tn = a + (n - 1)d, where d is the common difference.
• AP sum: Sn = n/2 × [2a + (n - 1)d] = n/2 × (first term + last term).
• Arithmetic mean of two numbers a and b = (a + b) ÷ 2; the means inserted between them form an AP.
• GP: nth term Tn = a × r^(n - 1), where r is the common ratio.
• GP sum: a(rⁿ - 1) ÷ (r - 1) and a(1 - rⁿ) ÷ (1 - r) are equal for any r ≠ 1. As a convenience to keep numbers positive, use the first when r > 1 and the second when |r| < 1.
• Sum of an infinite GP = a ÷ (1 - r), valid only when |r| < 1.
• Geometric mean of two positive numbers a and b = √(ab).
• HP: reciprocals of the terms form an AP, so convert to AP and solve.
• Harmonic mean of a and b = 2ab ÷ (a + b).
• For two positive numbers, AM × HM = GM², and AM ≥ GM ≥ HM, with equality only when the numbers are equal.
• Σn = n(n + 1) ÷ 2; Σn² = n(n + 1)(2n + 1) ÷ 6; Σn³ = [n(n + 1) ÷ 2]².
• Compound amount A = P(1 + r)ⁿ, with r as the rate per period; depreciation uses (1 - r)ⁿ.

Common mistakes

• Treating sequence and series as the same thing. Fix: Sequence is a list separated by commas. Series is the terms joined by plus signs, or their sum.
• Finding tₙ from Sₙ by subtracting S(n − 1) wrongly, such as writing Sₙ − 1. Fix: Replace every n in the formula for Sₙ with (n − 1), then subtract.
• Using n instead of (n − 1) in Tn = a + (n − 1)d. Fix: Test the formula on n = 1. You must get T1 = a. If not, the formula is wrong.
• Finding d by subtracting in the wrong order, giving the wrong sign. Fix: Always compute later term − earlier term. For 20, 17, 14, d = 17 − 20 = −3.
• Using d = (b − a) ÷ n instead of (b − a) ÷ (n + 1). Fix: Always say: n means, n + 2 terms, n + 1 gaps.
• Including a and b in the sum of the inserted means. Fix: The n means exclude a and b. Use n(a + b) ÷ 2, or subtract a + b from the total AP sum.
• Writing Tₙ = a·rⁿ instead of a·r^(n−1) Fix: Test with n = 1. The first term must be a, so the power must be 0. Hence n − 1.
• Finding r by subtracting terms Fix: In a GP always divide: r = T₂ ÷ T₁.
• Applying the AP formula directly to the HP terms. Fix: Write the reciprocals on the first line every time. Work only with them, then flip the result.
• Forgetting to take the final reciprocal. Fix: Label your answer as the term of the AP, then write 'HP term = 1 ÷ that value' before choosing an option.

Exam tips

• Expect direct questions: find a particular term from tₙ, or find tₙ from Sₙ. These are quick marks.
• Always check the first term when you are given Sₙ. Compute S₁ and compare.
• Read whether the question says 'term' or 'sum of terms'. Options often include both values.
• Questions on this topic are short. Do them first and save time for longer calculation questions.
• Sequence and series basics lead into AP and GP. Learn the notation tₙ and Sₙ well now.
• Look at the options first. If the data gives an easy a and d, the answer often comes in one line.
• Questions on 'sum of the first n natural, odd or even numbers' are standard AP sums. Use the shortcut: odd numbers sum to n², even numbers sum to n(n + 1).
• For word problems with equal yearly or monthly increases (salary rises, instalments), set up an AP and decide whether you need a term or a sum.