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Predicting What Comes Next: Exploring Sequences - Geometric Progressions

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 term by a fixed, non-zero constant called the common ratio rr.

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The first term of a GP is usually denoted by aa and the common ratio is denoted by rr.

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A sequence a1,a2,a3,…,ana_1, a_2, a_3, \dots, a_n is a GP if a2a1=a3a2=⋯=anan−1=r\frac{a_2}{a_1} = \frac{a_3}{a_2} = \dots = \frac{a_n}{a_{n-1}} = r.

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The general form of a GP is a,ar,ar2,ar3,…,arn−1a, ar, ar^2, ar^3, \dots, ar^{n-1}.

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If r>1r > 1, the terms of the GP increase (if a>0a > 0). If 0<r<10 < r < 1, the terms of the GP decrease.

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If rr is negative, the terms of the sequence will alternate between positive and negative values.

📐Formulae

an=a⋅rn−1a_n = a \cdot r^{n-1}

r=anan−1r = \frac{a_n}{a_{n-1}}

Sn=a(rn−1)r−1 (if r>1)S_n = \frac{a(r^n - 1)}{r - 1} \text{ (if } r > 1\text{)}

Sn=a(1−rn)1−r (if r<1)S_n = \frac{a(1 - r^n)}{1 - r} \text{ (if } r < 1\text{)}

Sn=n⋅a (if r=1)S_n = n \cdot a \text{ (if } r = 1\text{)}

💡Examples

Problem 1:

Find the 6th6^{th} term of the geometric progression 2,6,18,54,…2, 6, 18, 54, \dots.

Solution:

Given sequence: 2,6,18,54,…2, 6, 18, 54, \dots First term a=2a = 2 Common ratio r=62=3r = \frac{6}{2} = 3 We need to find a6a_6. Using the formula an=a⋅rn−1a_n = a \cdot r^{n-1}: a6=2⋅36−1a_6 = 2 \cdot 3^{6-1} a6=2⋅35a_6 = 2 \cdot 3^5 a6=2⋅243a_6 = 2 \cdot 243 a6=486a_6 = 486

Explanation:

To find a specific term in a GP, identify the first term and common ratio, then apply the general term formula.

Problem 2:

In a GP, the first term is 55 and the common ratio is 22. Find the sum of the first 55 terms.

Solution:

Given: a=5a = 5 r=2r = 2 n=5n = 5 Since r>1r > 1, we use the formula Sn=a(rn−1)r−1S_n = \frac{a(r^n - 1)}{r - 1}: S5=5(25−1)2−1S_5 = \frac{5(2^5 - 1)}{2 - 1} S5=5(32−1)1S_5 = \frac{5(32 - 1)}{1} S5=5⋅31S_5 = 5 \cdot 31 S5=155S_5 = 155

Explanation:

The sum of nn terms of a GP is calculated using the sum formula based on whether the common ratio rr is greater than or less than 11.

Problem 3:

Determine if the sequence 100,50,25,12.5100, 50, 25, 12.5 is a GP and find the common ratio.

Solution:

Check the ratios between consecutive terms: a2a1=50100=0.5\frac{a_2}{a_1} = \frac{50}{100} = 0.5 a3a2=2550=0.5\frac{a_3}{a_2} = \frac{25}{50} = 0.5 a4a3=12.525=0.5\frac{a_4}{a_3} = \frac{12.5}{25} = 0.5 Since the ratio is constant, the sequence is a GP with common ratio r=0.5r = 0.5 or r=12r = \frac{1}{2}.

Explanation:

A sequence is a GP if the ratio of any term to its preceding term remains constant throughout the sequence.