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Sequences and Progressions - Apply GP concepts in fractals and solve structured logic tasks like Tower of Hanoi

Grade 9CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A Geometric Progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio rr.

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In Fractals (like the Koch Snowflake or Sierpinski Triangle), geometric patterns are repeated at smaller scales. The number of new shapes added and the reduction in their size typically follow a GP.

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The Tower of Hanoi is a logical puzzle where the minimum number of moves required to transfer nn disks from one rod to another is given by the sequence 1,3,7,15,…1, 3, 7, 15, \dots, which is related to the sum of a GP where a=1a = 1 and r=2r = 2.

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For a GP with first term aa and common ratio rr, the nthn^{th} term is given by ana_n and the sum of the first nn terms is SnS_n.

📐Formulae

an=a⋅rn−1a_n = a \cdot r^{n-1}

r=an+1anr = \frac{a_{n+1}}{a_n}

Sn=a(rn−1)r−1 where r≠1S_n = \frac{a(r^n - 1)}{r - 1} \text{ where } r \neq 1

Mn=2n−1 (Minimum moves for Tower of Hanoi with n disks)M_n = 2^n - 1 \text{ (Minimum moves for Tower of Hanoi with } n \text{ disks)}

💡Examples

Problem 1:

In a fractal construction, the number of segments in each iteration follows a GP. If the first iteration has 44 segments and the second has 1616 segments, find the number of segments in the 5th5^{th} iteration.

Solution:

Given a1=4a_1 = 4 and a2=16a_2 = 16. First, find the common ratio rr: r=a2a1=164=4r = \frac{a_2}{a_1} = \frac{16}{4} = 4 Now, use the formula for the nthn^{th} term for n=5n=5: a5=a⋅r5−1a_5 = a \cdot r^{5-1} a5=4⋅44a_5 = 4 \cdot 4^4 a5=4⋅256=1024a_5 = 4 \cdot 256 = 1024

Explanation:

Since each iteration multiplies the number of segments by a constant factor, we identify it as a GP and solve for the specific term index.

Problem 2:

Calculate the minimum number of moves required to solve the Tower of Hanoi puzzle with 66 disks.

Solution:

The minimum number of moves is given by the formula: Mn=2n−1M_n = 2^n - 1 Substitute n=6n = 6: M6=26−1M_6 = 2^6 - 1 M6=64−1=63M_6 = 64 - 1 = 63

Explanation:

The Tower of Hanoi sequence 1,3,7,15...1, 3, 7, 15... is equivalent to SnS_n of a GP where a=1,r=2a=1, r=2, resulting in the formula 2n−12^n - 1.

Problem 3:

A square fractal starts with an area of 100 cm2100 \text{ cm}^2. In each subsequent step, the area added is exactly 12\frac{1}{2} of the area added in the previous step. Find the total area after 44 steps (including the initial square).

Solution:

This is a GP where a=100a = 100 and r=12r = \frac{1}{2}. We need the sum S4S_4: S4=a(1−r4)1−rS_4 = \frac{a(1 - r^4)}{1 - r} S4=100(1−(12)4)1−12S_4 = \frac{100(1 - (\frac{1}{2})^4)}{1 - \frac{1}{2}} S4=100(1−116)12S_4 = \frac{100(1 - \frac{1}{16})}{\frac{1}{2}} S4=100(1516)12=200×1516=300016=187.5 cm2S_4 = \frac{100(\frac{15}{16})}{\frac{1}{2}} = 200 \times \frac{15}{16} = \frac{3000}{16} = 187.5 \text{ cm}^2

Explanation:

The total area is the sum of the GP terms over 4 iterations.

Apply GP concepts in fractals and solve structured logic tasks like Tower of Hanoi Class 9 Notes &…