Calculating the Next Terms

Oct 13, 2025 by miodeya.com 10 views

When faced with a series of numbers, it's essential to identify the pattern and understand how each term is generated. In this case, we have a series of numbers: 2, 10, 50, 250, and so on. To determine the sum of the first eight terms in this series, we need to identify the pattern and then use it to calculate the sum.

Identifying the Pattern

The given series is: 2, 10, 50, 250, ...

To identify the pattern, let's examine the differences between consecutive terms:

10 - 2 = 8
50 - 10 = 40
250 - 50 = 200

We can see that the differences between consecutive terms are increasing by a factor of 5 each time (8 × 5 = 40, 40 × 5 = 200). This suggests that the series is formed by multiplying the previous term by 5 and then adding a certain value.

Calculating the Next Terms

To confirm our hypothesis, let's calculate the next few terms in the series:

250 × 5 = 1250 (add 8 to get the next term: 1250 + 8 = 1258)
1258 × 5 = 6290 (add 40 to get the next term: 6290 + 40 = 6330)
6330 × 5 = 31650 (add 200 to get the next term: 31650 + 200 = 31850)

Now that we have confirmed the pattern, we can use it to calculate the sum of the first eight terms.

Calculating the Sum

The first eight terms in the series are: 2, 10, 50, 250, 1258, 6330, 31650, 158250.

To calculate the sum, we can add these terms together:

2 + 10 + 50 + 250 + 1258 + 6330 + 31650 + 158250 = 195312

Conclusion

The sum of the first eight terms in the series is 195,312.

Answer

Q: What is the pattern in the given series?

A: The pattern in the given series is formed by multiplying the previous term by 5 and then adding a certain value. Specifically, the differences between consecutive terms are increasing by a factor of 5 each time.

Q: How do you calculate the next terms in the series?

A: To calculate the next terms in the series, we multiply the previous term by 5 and then add the next difference. For example, to get the next term after 250, we multiply 250 by 5 to get 1250, and then add 8 to get 1258.

Q: How do you calculate the sum of the first eight terms in the series?

A: To calculate the sum of the first eight terms in the series, we add the terms together: 2 + 10 + 50 + 250 + 1258 + 6330 + 31650 + 158250.

Q: What is the sum of the first eight terms in the series?

A: The sum of the first eight terms in the series is 195,312.

Q: Why is it essential to identify the pattern in the series?

A: Identifying the pattern in the series is essential because it allows us to calculate the next terms and the sum of the series. Without understanding the pattern, we would not be able to calculate the sum of the series.

Q: What are some common types of series?

A: Some common types of series include:

Arithmetic series: a series in which each term is obtained by adding a fixed constant to the previous term.
Geometric series: a series in which each term is obtained by multiplying the previous term by a fixed constant.
Harmonic series: a series in which each term is the reciprocal of a positive integer.

Q: How do you determine which type of series you are dealing with?

A: To determine which type of series you are dealing with, you need to examine the differences or ratios between consecutive terms. If the differences are constant, it is an arithmetic series. If the ratios are constant, it is a geometric series.

Q: What are some real-world applications of series?

A: Series have many real-world applications, including:

Finance: series are used to calculate interest rates and investments.
Science: series are used to model population growth and decay.
Engineering: series are used to calculate the stress and strain on materials.

Q: How do you use series in real-world applications?

A: To use series in real-world applications, you need to understand the pattern and calculate the sum of the series. For example, in finance, you can use series to calculate the future value of an investment.

Conclusion

Understanding series is essential in mathematics and has many real-world applications. By identifying the pattern in a series, we can calculate the sum of the series and use it to solve problems in finance, science, and engineering.