Finding The 50th Term And Constructing Arithmetic Sequences

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In the realm of arithmetic sequences, each term is generated by adding a constant value, commonly known as the common difference, to its preceding term. To fully grasp the intricacies of an arithmetic sequence, we need to decipher the fundamental formula that governs its behavior. The nth term, denoted as an, can be elegantly expressed as:

an = a1 + (n - 1)d

where a1 signifies the first term and d represents the common difference. This equation serves as the cornerstone for unraveling the mysteries of arithmetic sequences.

Now, let's delve into the problem at hand. We are presented with two crucial pieces of information: the 4th term is 2 and the 7th term is 3. Our mission is to determine the 50th term. To embark on this quest, we shall utilize the formula we've just unveiled. Substituting the given values, we arrive at two equations:

  1. a4 = a1 + 3d = 2
  2. a7 = a1 + 6d = 3

With this dynamic duo of equations in our arsenal, we can embark on a journey to solve for the unknowns, a1 and d. A particularly effective strategy involves the method of elimination. By subtracting the first equation from the second, we artfully eliminate a1, leaving us with:

3d = 1

Solving for d, we discover that the common difference is:

d = 1/3

Now that we've unearthed the common difference, we can substitute it back into either of our original equations to unveil the elusive first term, a1. Let's opt for the first equation:

a1 + 3(1/3) = 2

Simplifying, we find:

a1 + 1 = 2

Therefore, the first term is:

a1 = 1

Armed with both the first term and the common difference, we are now fully equipped to conquer our ultimate objective: finding the 50th term. Plugging n = 50 into our general formula, we obtain:

a50 = 1 + (50 - 1)(1/3)

a50 = 1 + 49/3

a50 = 1 + 16.333...

a50 = 17.333...

Thus, the 50th term of this arithmetic sequence is approximately 17.333. This value showcases the continuing progression along the sequence, propelled by the consistent addition of the common difference.

Now, let's transition to a different challenge: constructing an arithmetic sequence with a predefined first term and a specific sum for its initial terms. We are tasked with crafting a sequence where the first term (a1) is 30 and the sum of the first 5 terms (S5) equals 300. To tackle this task, we must invoke the formula for the sum of the first n terms of an arithmetic sequence:

Sn = (n/2)[2a1 + (n - 1)d]

In our scenario, we know that a1 = 30, n = 5, and S5 = 300. Substituting these values into the formula, we get:

300 = (5/2)[2(30) + (5 - 1)d]

Simplifying the equation, we have:

300 = (5/2)[60 + 4d]

To isolate the term with d, we can multiply both sides by 2/5:

120 = 60 + 4d

Subtracting 60 from both sides yields:

60 = 4d

Finally, dividing by 4, we unearth the common difference:

d = 15

With the first term (a1 = 30) and the common difference (d = 15) in our grasp, we can now explicitly define the arithmetic sequence. The sequence unfolds as follows:

30, 45, 60, 75, 90, ...

Each subsequent term is generated by adding 15 to the preceding term. This sequence showcases the harmonious interplay between the first term, the common difference, and the sum of the initial terms.

This exercise underscores the power of mathematical formulas in constructing and analyzing arithmetic sequences. By understanding the underlying principles, we can confidently navigate the world of sequences and series.

In this exploration, we've traversed two distinct facets of arithmetic sequences. First, we mastered the art of finding a specific term within a sequence, given partial information. We deftly applied the general formula for the nth term, showcasing its utility in solving problems involving arithmetic sequences. Second, we embarked on a constructive endeavor, fashioning an arithmetic sequence that adheres to predefined criteria. This endeavor highlighted the significance of the sum formula and its role in shaping sequences with desired properties.

Arithmetic sequences, with their inherent regularity and predictability, serve as a cornerstone in mathematics. Their applications extend far beyond the realm of textbooks, permeating fields such as physics, engineering, and computer science. By delving into the intricacies of arithmetic sequences, we not only enhance our mathematical prowess but also gain a deeper appreciation for the elegance and power of mathematical patterns.

Understanding arithmetic sequences is not just about memorizing formulas; it's about developing a logical and analytical approach to problem-solving. The ability to manipulate equations, identify patterns, and apply concepts is crucial for success in mathematics and beyond. The journey through these problems not only reinforces mathematical skills but also cultivates critical thinking, which is an indispensable asset in all facets of life. Therefore, the pursuit of mathematical knowledge is not merely an academic exercise; it's an investment in our intellectual growth and our capacity to navigate the complexities of the world around us.

Keywords

  • Arithmetic sequence
  • Common difference
  • First term
  • nth term
  • Sum of terms
  • 50th term
  • Sequence construction
  • Mathematical formulas
  • Problem-solving
  • Mathematical patterns