Type The Missing Number In This Sequence:1, 2, $\square$, 8, 16

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Introduction

Mathematics is a fascinating subject that involves the study of numbers, quantities, and shapes. It is a fundamental subject that has numerous applications in various fields, including science, technology, engineering, and mathematics (STEM). One of the most interesting aspects of mathematics is the study of sequences and patterns. In this article, we will explore a sequence of numbers and try to identify the missing number.

The Sequence

The given sequence is: 1, 2, $\square$, 8, 16. At first glance, the sequence appears to be a simple arithmetic progression, but it is not. The numbers in the sequence are not consecutive, and there is a pattern that needs to be identified.

Pattern Identification

To identify the pattern, let's examine the differences between consecutive numbers in the sequence.

  • 2 - 1 = 1

  • \square$ - 2 = ?

  • 8 - $\square$ = ?
  • 16 - 8 = 8

From the above calculations, we can see that the differences between consecutive numbers are not constant. However, there is a pattern that emerges when we look at the sequence in a different way.

Geometric Progression

Let's re-examine the sequence and look for a geometric progression. A geometric progression is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

  • 1 Γ— 2 = 2
  • 2 Γ— 4 = 8
  • 8 Γ— 2 = 16

From the above calculations, we can see that the sequence is a geometric progression with a common ratio of 4.

The Missing Number

Now that we have identified the pattern, we can easily find the missing number in the sequence.

  • 1 Γ— 2 = 2
  • 2 Γ— 4 = 8
  • 8 Γ· 4 = 2
  • 2 Γ— 4 = 8

Therefore, the missing number in the sequence is 4.

Conclusion

In conclusion, the missing number in the sequence 1, 2, $\square$, 8, 16 is 4. The sequence is a geometric progression with a common ratio of 4. This problem is a great example of how mathematics can be used to identify patterns and solve problems.

Real-World Applications

The concept of geometric progression has numerous real-world applications. For example, in finance, the concept of compound interest is based on geometric progression. When an amount is invested at a fixed interest rate, the interest earned in each period is added to the principal amount, resulting in a geometric progression.

Example 1: Compound Interest

Suppose an amount of $100 is invested at an interest rate of 10% per annum. The interest earned in the first year is $10, and the total amount becomes $110. In the second year, the interest earned is 10% of $110, which is $11. The total amount becomes $121. This process continues, resulting in a geometric progression.

Example 2: Population Growth

Suppose a population of 1000 people is growing at a rate of 10% per annum. The population in the first year is 1000, and in the second year, it becomes 1100. In the third year, the population becomes 1210, and so on. This process results in a geometric progression.

Final Thoughts

In conclusion, the missing number in the sequence 1, 2, $\square$, 8, 16 is 4. The sequence is a geometric progression with a common ratio of 4. This problem is a great example of how mathematics can be used to identify patterns and solve problems. The concept of geometric progression has numerous real-world applications, including finance and population growth.

References

Glossary

  • Geometric Progression: A sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
  • Common Ratio: A fixed, non-zero number that is used to find each term in a geometric progression.
  • Compound Interest: The interest earned on an investment over a period of time, resulting in a geometric progression.
  • Population Growth: The increase in the number of people in a population over a period of time, resulting in a geometric progression.
    Type the Missing Number in this Sequence: 1, 2, $\square$, 8, 16 - Q&A ===========================================================

Introduction

In our previous article, we explored a sequence of numbers and identified the missing number as 4. The sequence was a geometric progression with a common ratio of 4. In this article, we will answer some frequently asked questions related to the sequence and provide additional information to help you better understand the concept of geometric progression.

Q&A

Q: What is a geometric progression?

A: A geometric progression is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Q: What is the common ratio in the sequence 1, 2, $\square$, 8, 16?

A: The common ratio in the sequence 1, 2, $\square$, 8, 16 is 4.

Q: How do I find the missing number in a geometric progression?

A: To find the missing number in a geometric progression, you can use the formula:

an = ar^(n-1)

where an is the nth term, a is the first term, r is the common ratio, and n is the term number.

Q: What are some real-world applications of geometric progression?

A: Geometric progression has numerous real-world applications, including:

  • Compound interest in finance
  • Population growth in biology
  • Sales growth in business
  • Investment returns in finance

Q: How do I calculate compound interest?

A: To calculate compound interest, you can use the formula:

A = P(1 + r/n)^(nt)

where A is the amount of money accumulated after n years, including interest, P is the principal amount, r is the annual interest rate, n is the number of times that interest is compounded per year, and t is the time the money is invested for in years.

Q: What is the difference between arithmetic progression and geometric progression?

A: Arithmetic progression is a sequence of numbers in which each term after the first is found by adding a fixed, non-zero number called the common difference. Geometric progression, on the other hand, is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Q: How do I determine if a sequence is a geometric progression?

A: To determine if a sequence is a geometric progression, you can check if the ratio of consecutive terms is constant. If the ratio is constant, then the sequence is a geometric progression.

Conclusion

In conclusion, the missing number in the sequence 1, 2, $\square$, 8, 16 is 4. The sequence is a geometric progression with a common ratio of 4. We hope that this Q&A article has provided you with a better understanding of the concept of geometric progression and its real-world applications.

Additional Resources

Glossary

  • Geometric Progression: A sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
  • Common Ratio: A fixed, non-zero number that is used to find each term in a geometric progression.
  • Compound Interest: The interest earned on an investment over a period of time, resulting in a geometric progression.
  • Population Growth: The increase in the number of people in a population over a period of time, resulting in a geometric progression.