Arithmetic Progression and Geometric Progression: formula sheet

Key formulas

General term of a sequence

Tₙ = f(n)

A rule in n. Put in n = 1, 2, 3 to get the first, second, third term.

Term from a sum (series)

Tₙ = Sₙ − Sₙ₋₁ (for n ≥ 2), and T₁ = S₁

Sₙ is the sum of the first n terms. Use T₁ = S₁ separately for the first term.

Sum of the first n terms

Sₙ = T₁ + T₂ + … + Tₙ

This is the series made from the first n terms.

Sum of first n natural numbers

1 + 2 + 3 + … + n = n(n + 1) ÷ 2

Useful for checking series quickly.

Common difference

d = T2 − T1 = T3 − T2 = Tn − T(n−1)

Must be the same for every consecutive pair, or the sequence is not an AP.

nth term

Tn = a + (n − 1)d

a is the first term, n is the position of the term.

Number of terms

n = (l − a) ÷ d + 1

l is the last term. The answer must be a positive whole number.

Difference between any two terms

Tm − Tn = (m − n)d

Fastest way to find d when two terms are given.

Three terms in AP

2b = a + c

If a, b, c are in AP, the middle term is the average of the other two.

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)

Use when the first term a and last term l are known. Here l = a + (n − 1)d.

nth term from sums

Tn = Sn − S(n−1)

Valid for n ≥ 2. T1 = S1.

Arithmetic mean of two numbers

A = (x + y) ÷ 2

If x, A, y are in AP, then 2A = x + y.

Inserting n arithmetic means between a and b

d = (b − a) ÷ (n + 1)

The AP has n + 2 terms. The means are a + d, a + 2d, ..., a + nd.

Sum of the n inserted means

Sum of means = n × (a + b) ÷ 2

Equals n times the AM of a and b.

Three terms in AP

Take a − d, a, a + d

Their sum is 3a, so d cancels out when you add them.

Sum of first n natural numbers

1 + 2 + ... + n = n(n + 1) ÷ 2

A special case with a = 1 and d = 1.

Common ratio

r = T₂ ÷ T₁ = T₃ ÷ T₂ = Tₙ ÷ Tₙ₋₁

Divide a term by the one before it. All such ratios must be equal for a GP. The first term must not be zero.

nth term of a GP

Tₙ = a · rⁿ⁻¹

a is the first term, r the common ratio. The power is n − 1, not n.

Term from the end of a finite GP

nth term from the end = l · (1/r)ⁿ⁻¹

l is the last term. Treat the GP in reverse with ratio 1/r.

Relation between two terms

Tₘ ÷ Tₙ = r^(m − n)

Useful when two terms are given and a is not needed.

Three numbers in GP

a/r, a, ar

Product = a³. Choose this form when the product is given.

Condition for three numbers in GP

b² = ac

If a, b, c are in GP, then b is the middle term and b² = ac.

nth term of a GP

Tn = a × rⁿ⁻¹

Use it to find n or the last term. The power is n − 1, not n.

Sum of n terms (form for |r| > 1)

Sn = a(rⁿ − 1) ÷ (r − 1)

Valid for any r ≠ 1. Prefer this form when |r| > 1, for example r = 2 or r = −2, as it keeps the working simple.

Sum of n terms (form for |r| < 1)

Sn = a(1 − rⁿ) ÷ (1 − r)

Valid for any r ≠ 1 and gives the same value as the other form. Prefer it when |r| < 1, for example when r is a fraction, as it avoids negative signs.

Sum when r = 1

Sn = n × a

Every term equals a. The formulas above cannot be used because the denominator becomes zero.

Sum to infinity

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

If |r| ≥ 1, the infinite sum does not exist as a fixed number.

Common ratio

r = T2 ÷ T1 = T3 ÷ T2

Divide any term by the one before it. It can be negative or a fraction.

Geometric mean of two numbers

G = √(ab)

For positive a and b. G is the middle term of the GP a, G, b.

nth term of a GP

Tₙ = a × r^(n−1)

a is the first term and r is the common ratio.

Common ratio when inserting n means

r = (b ÷ a)^(1/(n+1))

The GP has n + 2 terms. The means are ar, ar², ..., arⁿ.

Product of n GMs inserted

G₁ × G₂ × ... × Gₙ = (ab)^(n/2)

G₁, G₂, ..., Gₙ are the n inserted means. The product equals Gⁿ, where G = √(ab) is the single GM of a and b. Useful for quick checks.

Three terms in GP

b² = ac

If a, b, c are in GP (all non-zero), the middle term squared equals the product of the outer two.

Arithmetic mean and harmonic mean of two numbers

AM = (a + b) ÷ 2; HM = 2ab ÷ (a + b)

HM is the reciprocal of the AM of the reciprocals.

Relation between AM, GM and HM

AM ≥ GM ≥ HM and GM² = AM × HM

For positive numbers. Equality holds only when a = b. The GM² relation is exact for two numbers.

Terms equidistant from the ends of a GP

T₁ × Tₙ = T₂ × Tₙ₋₁ = ...

Products of terms equally far from the two ends are equal.

nth term of AP

Tₙ = a + (n − 1)d

a is the first term, d is the fixed change. d is negative for a fixed decrease.

Sum of n terms of AP

Sₙ = n/2 × [2a + (n − 1)d] = n/2 × (a + l)

Use for total salary or total savings over n periods. l is the last term.

nth term of GP

Tₙ = a × rⁿ⁻¹

r = 1 + rate for growth, r = 1 − rate for decay.

Sum of n terms of GP

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

Valid when r ≠ 1. Use for total of amounts growing by a fixed percentage.

Value after n periods at a fixed percentage

Aₙ = A₀ × rⁿ

A₀ is the starting value, which is term 1. After n periods you are at the (n+1)th term.

Quick revision

• AP: each term is the previous term plus a constant d; d = T2 − T1.
• nth term of an AP: Tn = a + (n − 1)d.
• Sum of n terms of an AP: Sn = n/2 × [2a + (n − 1)d] = n/2 × (first term + last term).
• AM of two numbers a and b is (a + b) ÷ 2, and it is the middle term of an AP with those ends.
• GP: each term is the previous term times a constant r; r = T2 ÷ T1.
• nth term of a GP: Tn = arⁿ⁻¹.
• Sum of n terms of a GP: Sn = a(rⁿ − 1) ÷ (r − 1) for r ≠ 1; if r < 1 you may use a(1 − rⁿ) ÷ (1 − r).
• Sum to infinity of a GP: S∞ = a ÷ (1 − r), valid only when |r| < 1.
• GM of two positive numbers a and b is √(ab).
• For two positive numbers: AM ≥ GM ≥ HM, and GM² = AM × HM.
• Three numbers in AP can be taken as a − d, a, a + d; in GP as a/r, a, ar.
• Number of terms from first to last in an AP: n = (l − a) ÷ d + 1.

Common mistakes

• Treating sequence and series as the same thing. Fix: Sequence = list of terms. Series = sum of those terms. Ask yourself if the answer should be one term or a total.
• Using Tₙ = Sₙ − Sₙ₋₁ for n = 1. Fix: Always take T₁ = S₁. Use the subtraction only for n ≥ 2.
• Using Tn = a + nd instead of a + (n − 1)d. Fix: Check with n = 1. The formula must give T1 = a. Only (n − 1)d does this.
• Finding d as first term − second term. Fix: Always do later term − earlier term. For 20, 17, 14, d = 17 − 20 = −3.
• Using d = (b − a) ÷ n when inserting n means. Fix: Always divide by (n + 1). Count the gaps between a and b, not the means.
• Using the wrong n in Sn = n/2 × (a + l), for example by miscounting the terms. Fix: Find n from l = a + (n − 1)d first, then apply the sum formula.
• Using Tₙ = a·rⁿ instead of a·rⁿ⁻¹. Fix: Check with n = 1: the answer must be a. Only rⁿ⁻¹ gives that.
• Finding r by subtracting terms. Fix: For a GP always divide. For AP always subtract.
• Using a ÷ (1 − r) when |r| ≥ 1. Fix: Always check |r| < 1 first. If it fails, the infinite sum does not exist.
• Taking the wrong number of terms, using rⁿ⁻¹ instead of rⁿ in the sum formula. Fix: Remember: nth term has rⁿ⁻¹, but sum of n terms has rⁿ. Test with n = 1: the sum should equal a.

Exam tips

• Expect short direct questions: find a term, find a sum of a few terms, or state the difference between sequence and series.
• Substituting n is quick. Do it before trying to find any pattern.
• When Sₙ is given, always work out two sums with small numbers and subtract.
• Read whether the question wants a term or a sum. Options often include both values.
• There is no negative marking, so attempt every question even if you have to guess after eliminating options.
• Questions are objective, so use d = (Tm − Tn) ÷ (m − n) to skip the two-equation method and save time.
• Test the options in the formula when the question asks for a particular term or n. It is often faster than full algebra.
• Watch for the wording 'which term' versus 'what is the term'. One asks for n, the other for Tn.