Determine The Number Of Digits Used To Write All The Terms Of The Sequence: 7, 12, 17, 22, ..., 67.

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In the realm of mathematics, sequences hold a significant position, offering a structured way to explore patterns and relationships between numbers. Among the diverse types of sequences, arithmetic progressions stand out for their consistent nature, where each term is derived by adding a constant value to its predecessor. This article delves into the intricacies of determining the number of digits employed in writing all the terms of a specific arithmetic progression: 7, 12, 17, 22, ..., 67. We will embark on a step-by-step journey, unraveling the underlying concepts and techniques necessary to solve this problem effectively.

Understanding Arithmetic Progressions

Before we delve into the specifics of the problem, let's establish a firm grasp of arithmetic progressions. An arithmetic progression, also known as an arithmetic sequence, is a sequence of numbers where the difference between any two consecutive terms remains constant. This constant difference is referred to as the common difference, often denoted by the letter 'd'.

In our given sequence, 7, 12, 17, 22, ..., 67, we can readily identify that it is an arithmetic progression. The common difference, 'd', can be calculated by subtracting any term from its immediate successor. For instance, 12 - 7 = 5, 17 - 12 = 5, and so on. This confirms that the common difference is indeed 5.

To further solidify our understanding, let's introduce the general formula for the nth term of an arithmetic progression. If 'a' represents the first term and 'd' represents the common difference, then the nth term, denoted as an, can be expressed as:

an = a + (n - 1)d

This formula empowers us to determine any term in the sequence, provided we know the first term, the common difference, and the term's position in the sequence. For example, to find the 10th term of our sequence, we would substitute a = 7, d = 5, and n = 10 into the formula, yielding:

a10 = 7 + (10 - 1)5 = 7 + 45 = 52

Identifying the Number of Terms

Now that we have a solid understanding of arithmetic progressions, let's address the crucial task of identifying the number of terms in our given sequence: 7, 12, 17, 22, ..., 67. To accomplish this, we can leverage the general formula for the nth term that we introduced earlier.

We know that the last term in the sequence is 67. Let's denote the position of this term as 'n'. Therefore, we can set an = 67 and substitute the values of a = 7 and d = 5 into the formula:

67 = 7 + (n - 1)5

Now, we can solve this equation for 'n' to determine the number of terms:

67 = 7 + 5n - 5 67 = 2 + 5n 65 = 5n n = 13

This calculation reveals that there are 13 terms in the sequence. This is a crucial piece of information, as it lays the foundation for our next step: determining the number of digits used to write all the terms.

Calculating Digits for Each Term

With the number of terms in hand, we now shift our focus to calculating the number of digits required to write each term in the sequence. This involves examining the individual terms and their digit composition.

Let's list out the terms of the sequence and their corresponding number of digits:

  • 7 (1 digit)
  • 12 (2 digits)
  • 17 (2 digits)
  • 22 (2 digits)
  • 27 (2 digits)
  • 32 (2 digits)
  • 37 (2 digits)
  • 42 (2 digits)
  • 47 (2 digits)
  • 52 (2 digits)
  • 57 (2 digits)
  • 62 (2 digits)
  • 67 (2 digits)

From this list, we observe that the first term, 7, requires only 1 digit, while the remaining 12 terms each require 2 digits. This pattern is crucial for our next step: calculating the total number of digits.

Determining the Total Number of Digits

Finally, we arrive at the culmination of our journey: determining the total number of digits used to write all the terms of the sequence. This involves a simple calculation, combining the digit counts we established in the previous step.

We know that there is 1 term with 1 digit and 12 terms with 2 digits. Therefore, the total number of digits can be calculated as follows:

Total digits = (1 term * 1 digit/term) + (12 terms * 2 digits/term) Total digits = 1 + 24 Total digits = 25

Therefore, a total of 25 digits are used to write all the terms of the sequence 7, 12, 17, 22, ..., 67.

Conclusion

In this comprehensive exploration, we have successfully navigated the intricacies of determining the number of digits used to write all the terms of an arithmetic progression. We began by establishing a firm understanding of arithmetic progressions and their properties. We then identified the number of terms in the given sequence, followed by a meticulous calculation of the digits required for each term. Finally, we combined these findings to arrive at the total number of digits, showcasing the power of systematic problem-solving in mathematics.

This exercise not only provides a solution to a specific problem but also illuminates the broader principles of sequence analysis and digit manipulation. By mastering these concepts, you can confidently tackle similar challenges and deepen your appreciation for the elegance and precision of mathematics.

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FAQ

What is an arithmetic progression?

An arithmetic progression (AP), also known as an arithmetic sequence, is a sequence of numbers such that the difference between the consecutive terms is constant. This difference is called the common difference.

How do you find the number of terms in an arithmetic progression?

To find the number of terms in an arithmetic progression, you can use the formula: n = ((last term - first term) / common difference) + 1

How do you calculate the total number of digits used to write the terms of a sequence?

To calculate the total number of digits used to write the terms of a sequence, you first need to identify the number of digits in each term. Then, sum up the number of digits for all terms in the sequence.

Can this method be applied to other types of sequences?

While the general approach of counting digits applies to any sequence, the specific formulas and methods used in this article are tailored for arithmetic progressions. Other types of sequences, such as geometric progressions, may require different techniques.

Why is it important to understand arithmetic progressions?

Understanding arithmetic progressions is important because they are a fundamental concept in mathematics and have numerous applications in various fields, including finance, physics, and computer science. They also serve as a building block for more advanced mathematical concepts.

What is the formula for the nth term of an arithmetic progression?

The formula for the nth term (an) of an arithmetic progression is: an = a + (n - 1)d, where a is the first term, n is the term number, and d is the common difference.

How can I improve my problem-solving skills in mathematics?

To improve your problem-solving skills in mathematics, practice consistently, understand the underlying concepts, break down complex problems into smaller steps, and review your solutions to learn from mistakes.