What is the Sum of an Infinite GP?

S3-SA1-0428

Grade Level:

Class 7

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The sum of an infinite Geometric Progression (GP) is the total value you get when you add up all the terms of a GP that goes on forever. This sum only exists if the terms in the GP get smaller and smaller, approaching zero.

Simple Example

Quick Example

Imagine you have a magic samosa. You eat half of it (1/2). Then you eat half of what's left (1/4). Then half of that (1/8), and so on. Even though you keep eating, you will never eat more than the whole samosa. The 'sum' of all the parts you eat (1/2 + 1/4 + 1/8 + ...) approaches the value of 1 whole samosa.

Worked Example

Step-by-Step

Let's find the sum of the infinite GP: 5, 2.5, 1.25, 0.625, ...

Step 1: Identify the first term (a) and the common ratio (r).
First term (a) = 5

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Step 2: Calculate the common ratio (r) by dividing any term by its previous term.
r = 2.5 / 5 = 0.5
or r = 1.25 / 2.5 = 0.5

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Step 3: Check if the common ratio (r) is between -1 and 1 (i.e., |r| < 1). If not, the sum does not exist.
Here, r = 0.5, which is between -1 and 1. So, the sum exists.

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Step 4: Use the formula for the sum of an infinite GP: S = a / (1 - r).
S = 5 / (1 - 0.5)

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Step 5: Calculate the denominator.
S = 5 / 0.5

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Step 6: Perform the division.
S = 10

The sum of the infinite GP is 10.

Why It Matters

Understanding infinite sums helps scientists in fields like Physics predict how things behave over time, for example, the decay of radioactive materials. Data Scientists use similar concepts for analyzing complex data patterns. It even helps engineers design stable systems and understand how signals weaken over distance.

Common Mistakes

MISTAKE: Trying to find the sum of an infinite GP where the common ratio (r) is 2 or -3. | CORRECTION: The sum of an infinite GP only exists if the absolute value of the common ratio (|r|) is less than 1 (i.e., -1 < r < 1). If |r| >= 1, the sum goes to infinity and cannot be calculated.

MISTAKE: Confusing the formula for finite GP sum with infinite GP sum. | CORRECTION: The formula for an infinite GP sum is S = a / (1 - r). The formula for a finite GP sum is different and depends on the number of terms.

MISTAKE: Calculating the common ratio (r) incorrectly, for example, by subtracting terms instead of dividing. | CORRECTION: Always divide a term by its preceding term to find the common ratio (r). For example, r = (second term) / (first term).

Practice Questions

Try It Yourself

QUESTION: What is the sum of the infinite GP: 10, 5, 2.5, ...? | ANSWER: 20

QUESTION: If the first term of an infinite GP is 12 and the common ratio is 1/3, find its sum. | ANSWER: 18

QUESTION: An infinite GP has a first term of 8. If its sum is 16, what is the common ratio? | ANSWER: 0.5 or 1/2

MCQ

Quick Quiz

For which of the following common ratios (r) can the sum of an infinite GP be found?

r = 2

r = -1.5

r = 0.7

r = -1

The Correct Answer Is:

The sum of an infinite GP can only be found if the absolute value of the common ratio (|r|) is less than 1. Only 0.7 fits this condition, as |0.7| < 1.

Real World Connection

In the Real World

Imagine a bouncing cricket ball. Each bounce, it reaches a height that is a fraction of the previous bounce. For example, if it bounces to 0.8 times its previous height each time, the total distance it travels up and down before it stops is an infinite GP sum. This helps physicists understand energy loss in collisions.

Key Vocabulary

Key Terms

Geometric Progression (GP): 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. | Common Ratio (r): The constant factor by which each term in a GP is multiplied to get the next term. | Infinite GP: A Geometric Progression that has an endless number of terms. | First Term (a): The initial number in a Geometric Progression. | Sum to Infinity (S): The total value obtained by adding all terms of an infinite GP, if it exists.

What's Next

What to Learn Next

Great job understanding infinite GPs! Next, you can explore 'Applications of GP in Compound Interest and Depreciation'. This will show you how GPs are used to calculate money growth in banks or the value reduction of assets over time, making your learning even more practical!