Sequences and Series

Master the concepts of arithmetic, geometric, and other types of sequences with clear explanations and examples.

Simplified Quantitative Formulas: Progressions and Series

Sequence: List of numbers in order. Series = sum of sequence.
AP: Tₙ = a + (n–1)d. Sum: Sₙ = n/2 × [2a + (n–1)d].
Number of terms: N = (last–first)/d + 1.
AM: Average = (sum of terms)/n.
GP: Tₙ = a × r^(n–1). Sum: Sₙ = a(rⁿ–1)/(r–1). Infinite: S∞ = a/(1–r).
GM: n-th root of product. For GP, GM = middle term.
HP: Reciprocals form AP. n-th term = 1/Aₙ.
AGP: Terms are product of AP and GP. Sₙ = [a–(a+(n–1)d)rⁿ]/(1–r) + (dr(1–rⁿ⁻¹))/(1–r)².

Arithmetic Sequences

An arithmetic sequence is a sequence where the difference between consecutive terms is constant. This constant difference is called the common difference (d).

General Form

a₁, a₁+d, a₁+2d, a₁+3d, ...

a₁ is the first term
d is the common difference
nth term = a₁ + (n-1)d
Sum of n terms = n/2 × (2a₁ + (n-1)d)

Example

Find the sum of the first 20 terms of the arithmetic sequence: 5, 9, 13, ...

Solution:

First term (a₁) = 5
Common difference (d) = 9 - 5 = 4
Sum = n/2 × (2a₁ + (n-1)d)
Sum = 20/2 × (2×5 + (20-1)4)
Sum = 10 × (10 + 76) = 10 × 86 = 860

Geometric Sequences

A geometric sequence is a sequence where the ratio between consecutive terms is constant. This constant ratio is called the common ratio (r).

General Form

a₁, a₁r, a₁r², a₁r³, ...

a₁ is the first term
r is the common ratio
nth term = a₁r^(n-1)
Sum of n terms = a₁(1-r^n)/(1-r) when r ≠ 1
Sum of infinite terms = a₁/(1-r) when |r| < 1

Example 1

Find the 8th term of the geometric sequence: 2, 6, 18, 54, ...

Solution:

First term (a₁) = 2
Common ratio (r) = 6/2 = 3
nth term = a₁r^(n-1)
8th term = 2 × 3^(8-1) = 2 × 3^7 = 2 × 2187 = 4374

Example 2

Find the sum of first 6 terms of the geometric sequence: 4, 12, 36, 108, ...

Solution:

First term (a₁) = 4
Common ratio (r) = 12/4 = 3
Sum = a₁(1-r^n)/(1-r)
Sum = 4(1-3^6)/(1-3)
Sum = 4(1-729)/(-2) = 4(-728)/(-2) = 1456

Harmonic Sequences

A harmonic sequence is a sequence where the reciprocals of the terms form an arithmetic sequence.

General Form

1/a₁, 1/(a₁+d), 1/(a₁+2d), 1/(a₁+3d), ...

a₁ is the first term of the arithmetic sequence
d is the common difference
nth term = 1/(a₁ + (n-1)d)

Example

Find the 5th term of the harmonic sequence: 1, 1/2, 1/3, 1/4, ...

Solution:

The reciprocals form an arithmetic sequence: 1, 2, 3, 4, ...
First term (a₁) = 1
Common difference (d) = 1
5th term = 1/(a₁ + (5-1)d)
5th term = 1/(1 + 4) = 1/5

Fibonacci Sequence

The Fibonacci sequence is a sequence where each term is the sum of the two preceding terms.

General Form

F(n) = F(n-1) + F(n-2)

F(1) = 1
F(2) = 1
F(n) is the nth term

Example

Find the first 8 terms of the Fibonacci sequence

Solution:

F(1) = 1
F(2) = 1
F(3) = F(2) + F(1) = 1 + 1 = 2
F(4) = F(3) + F(2) = 2 + 1 = 3
F(5) = F(4) + F(3) = 3 + 2 = 5
F(6) = F(5) + F(4) = 5 + 3 = 8
F(7) = F(6) + F(5) = 8 + 5 = 13
F(8) = F(7) + F(6) = 13 + 8 = 21

Special Sequences

Some important special sequences and their properties.

Square Numbers

1, 4, 9, 16, 25, ...

nth term = n²

Sum of first n terms = n(n+1)(2n+1)/6

Cube Numbers

1, 8, 27, 64, 125, ...

nth term = n³

Sum of first n terms = [n(n+1)/2]²

Triangular Numbers

1, 3, 6, 10, 15, ...

nth term = n(n+1)/2

Sum of first n terms = n(n+1)(n+2)/6

Example 1

Find the sum of first 5 square numbers

Solution:

Sum = n(n+1)(2n+1)/6
Sum = 5(5+1)(2×5+1)/6
Sum = 5×6×11/6 = 55

Example 2

Find the 6th triangular number

Solution:

nth term = n(n+1)/2
6th term = 6(6+1)/2
6th term = 6×7/2 = 21

Practice Questions

Test your understanding of Sequences & Series with 20 fully solved, step-by-step questions designed for beginners.

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