Finding Terms And Nth Term From Sum Of First N Terms
In the fascinating world of mathematical sequences, understanding the relationship between the sum of the first n terms and the individual terms themselves is a crucial skill. This article delves into a specific scenario where the sum of the first n terms, denoted as S_n, is given by the formula S_n = (1/4)(5n^2 + 11n). Our primary goal is to unravel the intricacies of this sequence by determining its first three terms and deriving a general expression for the nth term. This exploration will not only enhance your understanding of sequences but also equip you with valuable problem-solving techniques applicable in various mathematical contexts.
1. Finding the First Three Terms
To find the first three terms of the sequence, we will leverage the provided formula for the sum of the first n terms, S_n = (1/4)(5n^2 + 11n). The fundamental principle we'll employ is that the kth term of a sequence, denoted as a_k, can be obtained by subtracting the sum of the first k-1 terms from the sum of the first k terms. Mathematically, this is expressed as a_k = S_k - S_(k-1). Let's apply this principle to calculate the first three terms.
Calculating the First Term (a_1)
The first term, a_1, is simply the sum of the first one term, which is S_1. Substituting n = 1 into the formula for S_n, we get:
S_1 = (1/4)(5(1)^2 + 11(1)) = (1/4)(5 + 11) = (1/4)(16) = 4
Therefore, the first term of the sequence, a_1, is 4. This initial term serves as the foundation upon which the rest of the sequence is built. It's the starting point, and understanding its value is crucial for comprehending the sequence's overall behavior.
Determining the Second Term (a_2)
To find the second term, a_2, we need to subtract the sum of the first one term (S_1) from the sum of the first two terms (S_2). First, let's calculate S_2 by substituting n = 2 into the formula for S_n:
S_2 = (1/4)(5(2)^2 + 11(2)) = (1/4)(5(4) + 22) = (1/4)(20 + 22) = (1/4)(42) = 10.5
Now, we can find a_2 using the formula a_2 = S_2 - S_1:
a_2 = 10.5 - 4 = 6.5
Thus, the second term of the sequence, a_2, is 6.5. This term provides further insight into the sequence's progression, revealing the difference between the first two terms and hinting at the underlying pattern.
Unveiling the Third Term (a_3)
Similarly, to find the third term, a_3, we subtract the sum of the first two terms (S_2) from the sum of the first three terms (S_3). Let's calculate S_3 by substituting n = 3 into the formula for S_n:
S_3 = (1/4)(5(3)^2 + 11(3)) = (1/4)(5(9) + 33) = (1/4)(45 + 33) = (1/4)(78) = 19.5
Now, we can find a_3 using the formula a_3 = S_3 - S_2:
a_3 = 19.5 - 10.5 = 9
Therefore, the third term of the sequence, a_3, is 9. By calculating the first three terms, we've established a foundation for understanding the sequence's behavior and can now attempt to identify a general pattern or formula for the nth term.
2. Finding the nth Term
Having determined the first three terms of the sequence, our next objective is to find the nth term, denoted as a_n. To achieve this, we will again employ the relationship between the sum of the first n terms (S_n) and the individual terms. As previously established, the nth term can be expressed as the difference between the sum of the first n terms and the sum of the first (n-1) terms: a_n = S_n - S_(n-1). This formula allows us to express any term in the sequence directly in terms of n, providing a powerful tool for analyzing and predicting the sequence's behavior.
Expressing S_(n-1)
Before we can apply the formula a_n = S_n - S_(n-1), we need to determine an expression for S_(n-1). This is achieved by substituting (n-1) in place of n in the given formula for S_n:
S_n = (1/4)(5n^2 + 11n)
Replacing n with (n-1), we get:
S_(n-1) = (1/4)(5(n-1)^2 + 11(n-1))
Now, let's simplify this expression by expanding and combining like terms:
S_(n-1) = (1/4)(5(n^2 - 2n + 1) + 11n - 11)
S_(n-1) = (1/4)(5n^2 - 10n + 5 + 11n - 11)
S_(n-1) = (1/4)(5n^2 + n - 6)
This simplified expression for S_(n-1) is crucial for calculating the nth term, as it allows us to directly substitute it into the formula a_n = S_n - S_(n-1).
Deriving the Formula for a_n
Now that we have expressions for both S_n and S_(n-1), we can substitute them into the formula a_n = S_n - S_(n-1) to find the nth term:
a_n = (1/4)(5n^2 + 11n) - (1/4)(5n^2 + n - 6)
To simplify this expression, we can factor out the (1/4) and combine the terms inside the parentheses:
a_n = (1/4)[(5n^2 + 11n) - (5n^2 + n - 6)]
a_n = (1/4)(5n^2 + 11n - 5n^2 - n + 6)
Now, we can combine like terms:
a_n = (1/4)(10n + 6)
Finally, we can simplify further by factoring out a 2 from the terms inside the parentheses:
a_n = (1/4)(2)(5n + 3)
a_n = (1/2)(5n + 3)
Therefore, the general formula for the nth term of the sequence is a_n = (1/2)(5n + 3)***. This formula is a powerful result, as it allows us to calculate any term in the sequence directly by simply substituting the desired value of n. It encapsulates the underlying pattern of the sequence and provides a concise representation of its behavior.
Verifying the Formula
To ensure the correctness of our derived formula for a_n, we can verify it by comparing its predictions with the first three terms we calculated earlier. Let's substitute n = 1, 2, and 3 into the formula and see if we obtain the same values for a_1, a_2, and a_3:
For n = 1:
a_1 = (1/2)(5(1) + 3) = (1/2)(5 + 3) = (1/2)(8) = 4
For n = 2:
a_2 = (1/2)(5(2) + 3) = (1/2)(10 + 3) = (1/2)(13) = 6.5
For n = 3:
a_3 = (1/2)(5(3) + 3) = (1/2)(15 + 3) = (1/2)(18) = 9
As we can see, the formula a_n = (1/2)(5n + 3) accurately predicts the first three terms of the sequence, confirming its validity. This verification step is crucial in ensuring the reliability of our result and provides confidence in the derived formula.
Conclusion
In this article, we embarked on a journey to unravel the intricacies of a sequence defined by the sum of its first n terms, S_n = (1/4)(5n^2 + 11n). We successfully determined the first three terms of the sequence and derived a general formula for the nth term, a_n = (1/2)(5n + 3)***. This exploration not only demonstrates the power of mathematical techniques in analyzing sequences but also highlights the importance of understanding the relationships between sums and individual terms. The methods and insights gained in this article can be applied to a wide range of sequence-related problems, enhancing your mathematical problem-solving abilities. Remember, the key to mastering sequences lies in understanding the underlying patterns and utilizing the appropriate formulas and techniques to unlock their secrets.
