Finding The 25th Term Of An Arithmetic Sequence
Embark on a mathematical journey to discover the 25th term of an arithmetic sequence! In this comprehensive exploration, we will dissect the fundamental concepts of arithmetic sequences, meticulously analyze the given sequence, and employ a step-by-step methodology to unveil the coveted 25th term. Whether you're a seasoned math aficionado or a curious novice, this guide will equip you with the knowledge and skills to conquer arithmetic sequences with unwavering confidence.
Delving into the Realm of Arithmetic Sequences
Arithmetic sequences, the bedrock of mathematical progressions, are characterized by a constant difference between consecutive terms. This steadfast difference, known as the common difference, serves as the sequence's defining attribute. To truly grasp the essence of arithmetic sequences, let's delve into their core components:
- Terms: The individual elements that constitute the sequence, each occupying a specific position.
- Common Difference: The unwavering value that separates successive terms, dictating the sequence's pattern.
- First Term: The sequence's genesis, the initial value from which the progression unfolds.
- nth Term: The term residing at the nth position within the sequence, often denoted as t_n.
Unraveling the Formula for the nth Term
The formula for the nth term of an arithmetic sequence serves as our mathematical compass, guiding us to any term within the sequence's expanse. This formula, a cornerstone of arithmetic sequence analysis, is expressed as:
- t_n = a + (n - 1)d
Where:
- t_n represents the nth term.
- a signifies the first term.
- n denotes the term's position within the sequence.
- d embodies the common difference.
This formula empowers us to calculate any term within the sequence, provided we possess the first term, common difference, and the desired term's position. It's a testament to the elegance and efficiency of mathematical expressions.
Deciphering the Given Arithmetic Sequence
Let's turn our attention to the arithmetic sequence at hand: 7, 21, 35, 49, and 63. Our mission is to determine the 25th term (t_25). To embark on this quest, we must first identify the sequence's defining characteristics: the first term and the common difference.
Identifying the First Term
The first term, the sequence's origin, is readily apparent: 7. This value serves as our anchor, the starting point from which the sequence unfurls.
Unveiling the Common Difference
The common difference, the sequence's unwavering rhythm, is the constant value that separates successive terms. To unveil this crucial value, we can subtract any term from its immediate successor. Let's subtract the first term (7) from the second term (21):
- 21 - 7 = 14
The common difference, therefore, is 14. This signifies that each term is 14 greater than its predecessor, establishing the sequence's consistent pattern.
Summarizing the Sequence's Characteristics
Having meticulously analyzed the sequence, we've unearthed its defining attributes:
- First term (a): 7
- Common difference (d): 14
With these values in our grasp, we're poised to calculate the 25th term, the object of our mathematical pursuit.
Unveiling the 25th Term: A Step-by-Step Approach
Now, the moment we've been building towards! Let's employ the formula for the nth term to calculate the 25th term (t_25) of the sequence. Recall the formula:
- t_n = a + (n - 1)d
We have:
- a = 7 (the first term)
- d = 14 (the common difference)
- n = 25 (the desired term's position)
Substituting these values into the formula, we obtain:
- t_25 = 7 + (25 - 1)14
Let's simplify this expression step by step:
- Parentheses First: (25 - 1) = 24
- Multiplication: 24 * 14 = 336
- Addition: 7 + 336 = 343
Therefore, t_25 = 343. The 25th term of the arithmetic sequence is 343!
The Grand Finale: The 25th Term Revealed
Through meticulous analysis and a step-by-step application of the nth term formula, we've successfully unveiled the 25th term of the arithmetic sequence. The value of t_25 is 343. This signifies that the 25th number in the sequence 7, 21, 35, 49, 63, ... is 343. Our mathematical quest has reached its triumphant conclusion.
Solidifying Understanding: Practice Problems
To solidify your understanding of arithmetic sequences and the nth term formula, let's tackle a couple of practice problems:
Practice Problem 1:
The first 5 terms of an arithmetic sequence are 3, 8, 13, 18, and 23. What is the value of t_20?
Solution:
- Identify the first term (a): 3
- Determine the common difference (d): 8 - 3 = 5
- Apply the formula for the nth term: t_n = a + (n - 1)d
- Substitute the values: t_20 = 3 + (20 - 1)5
- Simplify: t_20 = 3 + (19)5 = 3 + 95 = 98
Therefore, the 20th term (t_20) of the sequence is 98.
Practice Problem 2:
An arithmetic sequence has a first term of -5 and a common difference of 4. What is the value of t_15?
Solution:
- Identify the first term (a): -5
- Determine the common difference (d): 4
- Apply the formula for the nth term: t_n = a + (n - 1)d
- Substitute the values: t_15 = -5 + (15 - 1)4
- Simplify: t_15 = -5 + (14)4 = -5 + 56 = 51
Therefore, the 15th term (t_15) of the sequence is 51.
Mastering Arithmetic Sequences: Key Takeaways
As we conclude our exploration of arithmetic sequences, let's recap the key takeaways that will empower you to tackle any arithmetic sequence challenge:
- Arithmetic sequences are characterized by a constant difference between consecutive terms, known as the common difference.
- The formula for the nth term (t_n = a + (n - 1)d) is your mathematical compass, guiding you to any term within the sequence.
- Identify the first term (a) and the common difference (d) to unlock the sequence's secrets.
- Practice is the key to mastery. Work through various problems to solidify your understanding.
With these takeaways in your arsenal, you're well-equipped to conquer arithmetic sequences with confidence and precision.
Conclusion: A Journey of Mathematical Discovery
In this comprehensive exploration, we've embarked on a journey to unravel the intricacies of arithmetic sequences and unveil the 25th term of a specific sequence. From dissecting the fundamental concepts to applying the nth term formula, we've equipped ourselves with the tools to conquer these mathematical progressions.
The value of t_25, the 25th term of the sequence 7, 21, 35, 49, and 63, is 343. This numerical revelation stands as a testament to our analytical prowess and the power of mathematical formulas.
As you continue your mathematical pursuits, remember the key takeaways from this exploration. Embrace the elegance and efficiency of arithmetic sequences, and may your journey be filled with discoveries and triumphs.
