Exploring Sequences Finding The First Terms Of Successions

Hey guys! Let's dive into the fascinating world of sequences and successions! In mathematics, a sequence is simply an ordered list of numbers, and each number in the sequence is called a term. Successions are often defined by a general term, which is a formula that tells you how to find any term in the sequence based on its position. In this article, we'll explore how to write the first five or seven terms of successions defined by their general terms. We'll break down each example step-by-step, making it super easy to grasp the concepts. So, grab your calculators and let's get started!

1. Writing the First Five Terms of Successions

In this section, we'll tackle the task of finding the first five terms of various successions. We'll be working with different general terms, each with its unique formula. The key here is to understand how to substitute values into the general term to generate the terms of the sequence. Let's jump right in!

a) $a_n = 2n$

Our first sequence has the general term $a_n = 2n$. This formula tells us that to find the nth term, we simply multiply the position 'n' by 2. To find the first five terms, we'll substitute n = 1, 2, 3, 4, and 5 into the formula.

• Finding the first term ($a_1$):
  • Substitute n = 1: $a_1 = 2 * 1 = 2$.
  • So, the first term is 2.
• Finding the second term ($a_2$):
  • Substitute n = 2: $a_2 = 2 * 2 = 4$.
  • The second term is 4.
• Finding the third term ($a_3$):
  • Substitute n = 3: $a_3 = 2 * 3 = 6$.
  • The third term is 6.
• Finding the fourth term ($a_4$):
  • Substitute n = 4: $a_4 = 2 * 4 = 8$.
  • The fourth term is 8.
• Finding the fifth term ($a_5$):
  • Substitute n = 5: $a_5 = 2 * 5 = 10$.
  • The fifth term is 10.

Therefore, the first five terms of the sequence defined by $a_n = 2n$ are 2, 4, 6, 8, and 10. This is a simple arithmetic sequence where each term is 2 more than the previous term. Understanding how to derive these terms by substituting values into the general term is crucial. This fundamental skill forms the basis for understanding more complex sequences and series.

b) $a_n = 2n - 1$

Next up, we have the sequence defined by the general term $a_n = 2n - 1$. This formula is slightly different from the previous one, as we're now subtracting 1 from the result of multiplying 'n' by 2. Let's find the first five terms using the same method of substitution.

• Finding the first term ($a_1$):
  • Substitute n = 1: $a_1 = 2 * 1 - 1 = 1$.
  • The first term is 1.
• Finding the second term ($a_2$):
  • Substitute n = 2: $a_2 = 2 * 2 - 1 = 3$.
  • The second term is 3.
• Finding the third term ($a_3$):
  • Substitute n = 3: $a_3 = 2 * 3 - 1 = 5$.
  • The third term is 5.
• Finding the fourth term ($a_4$):
  • Substitute n = 4: $a_4 = 2 * 4 - 1 = 7$.
  • The fourth term is 7.
• Finding the fifth term ($a_5$):
  • Substitute n = 5: $a_5 = 2 * 5 - 1 = 9$.
  • The fifth term is 9.

So, the first five terms of the sequence defined by $a_n = 2n - 1$ are 1, 3, 5, 7, and 9. This is another arithmetic sequence, but this time, the terms are the odd positive integers. The key takeaway here is to carefully follow the order of operations when substituting values into the general term. Multiply before you subtract!

c) $a_n = n(n-1)$

Now, let's look at a sequence with a slightly more complex general term: $a_n = n(n-1)$. This formula involves multiplying 'n' by (n-1). We'll follow the same procedure to find the first five terms.

• Finding the first term ($a_1$):
  • Substitute n = 1: $a_1 = 1 * (1 - 1) = 1 * 0 = 0$.
  • The first term is 0.
• Finding the second term ($a_2$):
  • Substitute n = 2: $a_2 = 2 * (2 - 1) = 2 * 1 = 2$.
  • The second term is 2.
• Finding the third term ($a_3$):
  • Substitute n = 3: $a_3 = 3 * (3 - 1) = 3 * 2 = 6$.
  • The third term is 6.
• Finding the fourth term ($a_4$):
  • Substitute n = 4: $a_4 = 4 * (4 - 1) = 4 * 3 = 12$.
  • The fourth term is 12.
• Finding the fifth term ($a_5$):
  • Substitute n = 5: $a_5 = 5 * (5 - 1) = 5 * 4 = 20$.
  • The fifth term is 20.

Therefore, the first five terms of the sequence defined by $a_n = n(n-1)$ are 0, 2, 6, 12, and 20. This sequence increases at an increasing rate, demonstrating how different general terms can lead to very different patterns. Pay close attention to the parentheses in the formula and remember to perform the subtraction inside the parentheses before multiplying.

d) $a_n = \frac{n}{n^2 + 1}$

This time, we have a general term that involves a fraction: $a_n = \frac{n}{n^2 + 1}$. This means we'll be dividing 'n' by the result of $n^2 + 1$. Let's find the first five terms, carefully following the order of operations.

• Finding the first term ($a_1$):
  • Substitute n = 1: $a_1 = \frac{1}{1^2 + 1} = \frac{1}{1 + 1} = \frac{1}{2}$.
  • The first term is 1/2.
• Finding the second term ($a_2$):
  • Substitute n = 2: $a_2 = \frac{2}{2^2 + 1} = \frac{2}{4 + 1} = \frac{2}{5}$.
  • The second term is 2/5.
• Finding the third term ($a_3$):
  • Substitute n = 3: $a_3 = \frac{3}{3^2 + 1} = \frac{3}{9 + 1} = \frac{3}{10}$.
  • The third term is 3/10.
• Finding the fourth term ($a_4$):
  • Substitute n = 4: $a_4 = \frac{4}{4^2 + 1} = \frac{4}{16 + 1} = \frac{4}{17}$.
  • The fourth term is 4/17.
• Finding the fifth term ($a_5$):
  • Substitute n = 5: $a_5 = \frac{5}{5^2 + 1} = \frac{5}{25 + 1} = \frac{5}{26}$.
  • The fifth term is 5/26.

Therefore, the first five terms of the sequence defined by $a_n = \frac{n}{n^2 + 1}$ are 1/2, 2/5, 3/10, 4/17, and 5/26. When dealing with fractions, remember to simplify if possible. In this case, none of the fractions can be simplified further. Also, remember to calculate the denominator ($n^2 + 1$) before performing the division.

e) $a_n = \frac{1 - (-1)^n}{3^n}$

Our final example for finding the first five terms involves a general term with both an exponent and a negative sign: $a_n = \frac{1 - (-1)^n}{3^n}$. This formula might look a bit intimidating, but don't worry! We'll break it down step-by-step. The key here is to remember the rules for exponents and negative numbers.

• Finding the first term ($a_1$):
  • Substitute n = 1: $a_1 = \frac{1 - (-1)^1}{3^1} = \frac{1 - (-1)}{3} = \frac{1 + 1}{3} = \frac{2}{3}$.
  • The first term is 2/3.
• Finding the second term ($a_2$):
  • Substitute n = 2: $a_2 = \frac{1 - (-1)^2}{3^2} = \frac{1 - 1}{9} = \frac{0}{9} = 0$.
  • The second term is 0.
• Finding the third term ($a_3$):
  • Substitute n = 3: $a_3 = \frac{1 - (-1)^3}{3^3} = \frac{1 - (-1)}{27} = \frac{1 + 1}{27} = \frac{2}{27}$.
  • The third term is 2/27.
• Finding the fourth term ($a_4$):
  • Substitute n = 4: $a_4 = \frac{1 - (-1)^4}{3^4} = \frac{1 - 1}{81} = \frac{0}{81} = 0$.
  • The fourth term is 0.
• Finding the fifth term ($a_5$):
  • Substitute n = 5: $a_5 = \frac{1 - (-1)^5}{3^5} = \frac{1 - (-1)}{243} = \frac{1 + 1}{243} = \frac{2}{243}$.
  • The fifth term is 2/243.

Thus, the first five terms of the sequence defined by $a_n = \frac{1 - (-1)^n}{3^n}$ are 2/3, 0, 2/27, 0, and 2/243. Notice how the terms alternate between non-zero values and zero. This is due to the $(-1)^n$ part of the formula, which alternates between -1 and 1 depending on whether 'n' is odd or even. Remember, a negative number raised to an even power is positive, and a negative number raised to an odd power is negative. This example highlights the importance of understanding the behavior of exponents and negative signs in mathematical expressions.

2. Determining the First Seven Terms of a Succession

Now, let's move on to finding the first seven terms of a succession. We'll be applying the same principles we used earlier, but this time, we'll extend our calculations to include the sixth and seventh terms. This will give us a more complete picture of the sequence's behavior.

Unfortunately, the provided question only states "Determine 7 primeiros termos da sucessão definida" without providing the general term for the succession. To proceed, we need the formula that defines the sequence. Without the general term ($a_n = ?$), we cannot calculate the first seven terms.

However, let's illustrate how we would find the first seven terms if we had a general term. We'll use the example from section 1a, $a_n = 2n$, and extend it to seven terms.

• We already know the first five terms: 2, 4, 6, 8, 10.
• Finding the sixth term ($a_6$):
  • Substitute n = 6: $a_6 = 2 * 6 = 12$.
  • The sixth term is 12.
• Finding the seventh term ($a_7$):
  • Substitute n = 7: $a_7 = 2 * 7 = 14$.
  • The seventh term is 14.

Therefore, the first seven terms of the sequence defined by $a_n = 2n$ are 2, 4, 6, 8, 10, 12, and 14.

This example demonstrates the straightforward process of extending our calculations to find more terms in a sequence. The key is to consistently apply the general term formula, substituting the appropriate value of 'n' for each term you want to find.

To solve the original question, please provide the general term of the succession. Once we have the formula, we can easily calculate the first seven terms.

Conclusion

In this article, we've explored how to write the first few terms of successions defined by their general terms. We've worked through several examples, each with a unique formula, and we've seen how to substitute values into these formulas to generate the terms of the sequence. Remember, the key is to carefully follow the order of operations and pay attention to any exponents, negative signs, or fractions in the general term. Understanding sequences is a fundamental concept in mathematics, and mastering this skill will set you up for success in more advanced topics. Keep practicing, guys, and you'll become sequence pros in no time! Remember, providing the general term is crucial for determining any sequence. So, next time you encounter a sequence problem, make sure you have the formula handy! Happy calculating!