Magical Math Sequences

(i) 8, 24, 72, ...

Let's examine the pattern:

8 × 3 = 24

24 × 3 = 72

This is a geometric sequence where each term is multiplied by 3.

Next terms: 72 × 3 = 216

216 × 3 = 648

648 × 3 = 1,944

(ii) 5, 1, -3, ...

Let's look at the differences:

1 - 5 = -4

-3 - 1 = -4

This is an arithmetic sequence with common difference -4.

Next terms: -3 + (-4) = -7

-7 + (-4) = -11

-11 + (-4) = -15

(iii) 1/4, 2/9, 3/16, ...

Let's analyze the pattern:

Numerators: 1, 2, 3,... (increasing by 1)

Denominators: 4, 9, 16,... (perfect squares: 2², 3², 4²)

So the nth term is n/(n+1)²

Next terms:

4/5² = 4/25

5/6² = 5/36

6/7² = 6/49

(i) aₙ = 3n - 2

Let's find terms for n = 1, 2, 3, 4:

a₁ = 3(1) - 2 = 1

a₂ = 3(2) - 2 = 4

a₃ = 3(3) - 2 = 7

a₄ = 3(4) - 2 = 10

Sequence: 1, 4, 7, 10,...

(ii) aₙ = (-1)ⁿ⁺¹ n(n+1)

Let's calculate each term:

a₁ = (-1)² × 1 × 2 = 1 × 2 = 2

a₂ = (-1)³ × 2 × 3 = -1 × 6 = -6

a₃ = (-1)⁴ × 3 × 4 = 1 × 12 = 12

a₄ = (-1)⁵ × 4 × 5 = -1 × 20 = -20

Sequence: 2, -6, 12, -20,...

(iii) aₙ = 2n² - 6

Calculating each term:

a₁ = 2(1)² - 6 = 2 - 6 = -4

a₂ = 2(2)² - 6 = 8 - 6 = 2

a₃ = 2(3)² - 6 = 18 - 6 = 12

a₄ = 2(4)² - 6 = 32 - 6 = 26

Sequence: -4, 2, 12, 26,...

(i) 2, 5, 10, 17,...

Let's examine the pattern:

2 = 1² + 1

5 = 2² + 1

10 = 3² + 1

17 = 4² + 1

This suggests aₙ = n² + 1

Let's verify:

a₅ = 5² + 1 = 26 (next term would be 26)

So nth term: aₙ = n² + 1

(ii) 0, 1/2, 2/3, 3/4,...

Looking at numerators and denominators:

Numerators: 0, 1, 2, 3,... (n-1)

Denominators: 1, 2, 3, 4,... (n)

So nth term: aₙ = (n-1)/n

Let's verify:

a₅ = (5-1)/5 = 4/5 (next term would be 4/5)

(iii) 3, 8, 13, 18,...

This is an arithmetic sequence:

First term (a₁) = 3

Common difference (d) = 8 - 3 = 5

General form: aₙ = a₁ + (n-1)d

So nth term: aₙ = 3 + (n-1)×5 = 5n - 2

Let's verify:

a₅ = 5×5 - 2 = 23 (next term would be 23)

(i) aₙ = 5n/(n+2)

Calculating a₆:

a₆ = 5×6 / (6+2) = 30/8 = 15/4

Calculating a₁₃:

a₁₃ = 5×13 / (13+2) = 65/15 = 13/3

(ii) aₙ = 2ⁿ - n

Calculating a₄:

a₄ = 2⁴ - 4 = 16 - 4 = 12

Calculating a₁₁:

a₁₁ = 2¹¹ - 11 = 2048 - 11 = 2037

Finding a₈ (n=8, which is even):

Using the even formula: aₙ = (n²-2)/n

a₈ = (8² - 2)/8 = (64 - 2)/8 = 62/8 = 31/4

Finding a₁₅ (n=15, which is odd):

Using the odd formula: aₙ = (2n+1)/3

a₁₅ = (2×15 + 1)/3 = (30 + 1)/3 = 31/3 = 31/3

Given:

a₁ = 1

a₂ = 1

Calculating subsequent terms:

a₃ = a₁ + a₂ = 1 + 1 = 2

a₄ = a₂ + a₃ = 1 + 2 = 3

a₅ = a₃ + a₄ = 2 + 3 = 5

a₆ = a₄ + a₅ = 3 + 5 = 8

First six terms:

1, 1, 2, 3, 5, 8

This is the famous Fibonacci sequence!