Lesson 10 of 10 · 15 min

Chapter review

Watch a class

The whole chapter on YouTube

AP, GP and means with practice questions
JEE Wallah · Hinglish · Whole chapter · Open on YouTube

Progressions and special series for JEE Main
Vedantu JEE Made Ejee · Hinglish · Whole chapter · Open on YouTube

Full NCERT explanation of sequences and series
Science and Fun Education · Hinglish · Whole chapter · Open on YouTube

Must-know facts

18 facts

1A sequence is an ordered list a₁, a₂, …; aₙ is its nth or general term.
2A finite sequence has a fixed number of terms; an infinite one never ends.
3A sequence is a function on the natural numbers.
4A series is the indicated sum a₁ + a₂ + …; Σ aₖ (k = 1 to n) is its sigma form.
5G.P.: non-zero terms with aₖ₊₁/aₖ = r constant.
6nth term of a G.P.: aₙ = arⁿ⁻¹.
7Sum of n terms: Sₙ = a(rⁿ − 1)/(r − 1) = a(1 − rⁿ)/(1 − r) for r ≠ 1; Sₙ = na for r = 1.
8The sum formula comes from subtracting rSₙ from Sₙ.
910 generations of ancestors: 2 + 4 + … + 1024 = 2046.
10Three terms in G.P.: take a/r, a, ar.
11G.M. of positive a and b: √(ab); G.M. of 2 and 8 is 4.
12Inserting n G.M.s between a and b: r = (b/a)^(1/(n + 1)).
13Between 1 and 256 the three G.M.s are 4, 16, 64.
14A.M. = (a + b)/2; A − G = (√a − √b)²/2 ≥ 0, so A ≥ G.
15A = G only when a = b.
16Numbers with A.M. A and G.M. G are the roots of x² − 2Ax + G² = 0.
17A.M. 10 and G.M. 8 give the numbers 4 and 16.
18Fibonacci: a₁ = a₂ = 1, aₙ = aₙ₋₁ + aₙ₋₂.

Common traps

Where marks are lost

Writing the nth term of a G.P. as arⁿ.

The first term is ar⁰, so the nth term is arⁿ⁻¹. a₁₀ of 3, 6, 12, … is 3 × 2⁹ = 1536.

Using Sₙ = a(rⁿ − 1)/(r − 1) when r = 1.

It divides by zero; when r = 1 every term is a and Sₙ = na.

Calling 0, 0, 0, … or 2, 0, 0, … a G.P.

A G.P. needs every term non-zero, so that each ratio aₖ₊₁/aₖ exists.

Taking r as the first term divided by the second.

r is a later term divided by the one before it: in 81, −27, 9, …, r = −27/81 = −1/3.

Treating 5 + 55 + 555 + … as a G.P. with r = 11.

The ratios 11 and 10.09… are not equal. Write each term as (5/9)(10ᵏ − 1) and split into a G.P. and a constant sum.

Forgetting the negative root when inserting an odd count of G.M.s.

r⁴ = 81 gives r = 3 or r = −3 (real roots); r = −3 gives −6, 18, −54.

Using A ≥ G for numbers that may be negative.

The result needs a, b > 0, since √(ab) and √a, √b must be real.

Confusing the series (the indicated sum) with the sum of the series (a number).

1 + 3 + 5 + 7 + 9 is a series; 25 is its sum.

Formulas

11 to know

General term of a G.P.

aₙ = arⁿ⁻¹

a first term, r common ratio.

Common ratio

r = aₖ₊₁ / aₖ

The same for every k; all terms non-zero.

Sum of n terms (r > 1)

Sₙ = a(rⁿ − 1)/(r − 1)

Valid for any r ≠ 1.

Sum of n terms (|r| < 1)

Sₙ = a(1 − rⁿ)/(1 − r)

Same formula, signs arranged to stay positive.

Sum when r = 1

Sₙ = na

All terms equal a.

Three terms in G.P.

a/r, a, ar

Product a³.

Geometric mean

G = √(ab)

a, b > 0.

n inserted G.M.s

r = (b/a)^(1/(n + 1)), Gₖ = arᵏ

b is the (n + 2)th term.

Arithmetic mean

A = (a + b)/2

A.M.–G.M. inequality

A − G = (√a − √b)²/2 ≥ 0

Equality only when a = b.

Numbers from A and G

x² − 2Ax + G² = 0

Its roots are the two numbers.

Key terms

13 terms

Sequence: Numbers listed in a definite order, with a first, second, third term and so on.
Term: One number of a sequence; aₙ is the term in position n.
General term: The nth term aₙ, written as a rule in n.
Finite sequence: A sequence with a fixed number of terms.
Infinite sequence: A sequence that does not end.
Recurrence relation: A rule giving each term from earlier terms, such as aₙ = aₙ₋₁ + aₙ₋₂.
Series: The indicated sum a₁ + a₂ + a₃ + … of the terms of a sequence.
Sigma notation: Σ aₖ with limits, a short way of writing a series.
Progression: A sequence whose terms follow a definite pattern.
Geometric progression: A sequence of non-zero terms with a constant ratio between consecutive terms.
Common ratio: The constant r = aₖ₊₁/aₖ of a G.P.
Geometric mean: √(ab) for positive a and b; more generally, numbers inserted to make a G.P.
Arithmetic mean: (a + b)/2, the number halfway between a and b.