Unit 4: Progressions
Mathematics notes · PTU syllabus (UGCC2501)
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Unit summary
A progression is a sequence of numbers that follows a fixed rule. Progressions model regular growth: equal instalments, interest, depreciation and repeating patterns in algorithms.
This unit covers arithmetic progressions (adding a fixed number each time) and geometric progressions (multiplying by a fixed number each time), with their nth terms, sums and means.
After this unit you can
- Recognise an AP and a GP
- Find the nth term and the sum of n terms of an AP
- Insert arithmetic means between two numbers
- Find the nth term of a GP and the geometric mean of two numbers
PTU syllabus topics
- Introduction
- arithmetic progression
- sum of a finite number of quantities in AP
- arithmetic means
- geometric progression
- geometric mean
nth term of AP
aₙ = a + (n − 1)d
Sum of n terms of AP
Sₙ = n/2 [2a + (n − 1)d]
nth term of GP
aₙ = a·rⁿ⁻¹
Sum of n terms of GP
Sₙ = a(rⁿ − 1)/(r − 1), r ≠ 1
Arithmetic mean of a, b
(a + b) / 2
Geometric mean of a, b
√(ab)
Topic 1
Introduction to sequences and progressions
A sequence is an ordered list of numbers called terms: a₁, a₂, a₃, … A progression is a sequence whose terms follow a definite rule.
- 3, 7, 11, 15, … — add 4 each time (arithmetic).
- 2, 6, 18, 54, … — multiply by 3 each time (geometric).
Rule
Add a fixed common difference d
Multiply by a fixed common ratio r
nth term
a + (n − 1)d
a·rⁿ⁻¹
Example
5, 8, 11, 14
3, 6, 12, 24
Mean of two numbers a and b
(a + b) / 2
√(ab)
Topic 2
Arithmetic progression (AP)
A sequence is an AP if the difference between consecutive terms is constant. This constant is the common difference d = a₂ − a₁.
- nth term: aₙ = a + (n − 1)d, where a is the first term.
Example
Find the 20th term of 7, 11, 15, … Here a = 7, d = 4. a₂₀ = 7 + 19 × 4 = 83.
Example
Which term of 3, 8, 13, … is 98? 98 = 3 + (n − 1)5, so n − 1 = 19 and n = 20.
Exam tip
To test whether a sequence is an AP, check that a₂ − a₁ = a₃ − a₂ = … and state the value of d.
Topic 3
Sum of n terms of an AP
The sum of the first n terms is
- Sₙ = n/2 [2a + (n − 1)d], or
- Sₙ = n/2 (a + l), where l is the last term.
Example
Sum of the first 50 natural numbers: a = 1, l = 50, n = 50. S = 50/2 × (1 + 50) = 1275.
Example
Sum of 10 terms of 2, 5, 8, …: S₁₀ = 10/2 [2 × 2 + 9 × 3] = 5 × 31 = 155.
Topic 4
Arithmetic means
If a, A, b are in AP, then A is the arithmetic mean (AM) of a and b: A = (a + b) / 2. To insert n arithmetic means between a and b, the full sequence has n + 2 terms, so d = (b − a) / (n + 1).
Example
Insert 3 AMs between 4 and 20. d = (20 − 4)/4 = 4, so the means are 8, 12, 16.
Topic 5
Geometric progression (GP) and geometric mean
A sequence is a GP if the ratio of consecutive terms is constant. This is the common ratio r = a₂ / a₁.
- nth term: aₙ = a·rⁿ⁻¹
- Sum of n terms: Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1
- Geometric mean (GM) of two positive numbers a and b: G = √(ab)
Example
The 6th term of 2, 6, 18, … is 2 × 3⁵ = 486.
Example
GM of 4 and 16 = √64 = 8, and 4, 8, 16 is a GP.
For two positive numbers, AM ≥ GM: for 4 and 16, AM = 10 and GM = 8.
Key terms
- Sequence
- An ordered list of numbers
- Common difference
- The fixed amount added in an AP, d = a₂ − a₁
- Common ratio
- The fixed multiplier in a GP, r = a₂ / a₁
- Arithmetic mean
- (a + b) / 2
- Geometric mean
- √(ab)
Quick revision
- AP: aₙ = a + (n − 1)d; Sₙ = n/2 [2a + (n − 1)d].
- To insert n AMs: d = (b − a)/(n + 1).
- GP: aₙ = a·rⁿ⁻¹; Sₙ = a(rⁿ − 1)/(r − 1).
- GM of a and b is √(ab); AM ≥ GM for positive numbers.
Important exam questions
Practice questions written to the PTU exam pattern for this unit's syllabus: short answers (Section A style) and long answers (Sections B and C style).
Short-answer questions
- Q1.Define an arithmetic progression with an example.
- Q2.Find the 15th term of the AP 5, 9, 13, …
- Q3.Find the arithmetic mean of 12 and 30.
- Q4.Define a geometric progression and its common ratio.
- Q5.Find the geometric mean of 9 and 25.
- Q6.Write the formula for the sum of n terms of an AP.
Long-answer questions
- Q1.Derive the formula for the sum of n terms of an AP and find the sum of the first 40 odd numbers.
- Q2.Insert five arithmetic means between 3 and 27.
- Q3.The 5th term of an AP is 30 and the 12th term is 65. Find the AP and the sum of its first 20 terms.
- Q4.Find the 8th term and the sum of the first 6 terms of the GP 3, 6, 12, … and find the GM of its 2nd and 4th terms.
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