Lesson 10 of 10 · 15 min
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Progressions and special series for JEE Main
Full NCERT explanation of sequences and series
Must-know facts
18 facts
- 01A sequence is an ordered list a₁, a₂, …; aₙ is its nth or general term.
- 02A finite sequence has a fixed number of terms; an infinite one never ends.
- 03A sequence is a function on the natural numbers.
- 04A series is the indicated sum a₁ + a₂ + …; Σ aₖ (k = 1 to n) is its sigma form.
- 05G.P.: non-zero terms with aₖ₊₁/aₖ = r constant.
- 06nth term of a G.P.: aₙ = arⁿ⁻¹.
- 07Sum of n terms: Sₙ = a(rⁿ − 1)/(r − 1) = a(1 − rⁿ)/(1 − r) for r ≠ 1; Sₙ = na for r = 1.
- 08The sum formula comes from subtracting rSₙ from Sₙ.
- 0910 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
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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.