Problem 114 - Precalculus with Limits

Question

The sum of the first \( n \) terms of anarithmetic sequence with first term \( a_1 \) and common difference \( d \) is \( S_n \) Determine the sum if each term is increased by \( 5 \). Explain

Step-by-Step Solution

Answer

After each term in the sequence is increased by 5, the new sum \(S_n'\) of the sequence will be \(S_n' = S_n + 5n\), where \(S_n\) is the sum of the original sequence and \(n\) represents the number of terms in the sequence.

Step 1 - Analyzing New Sequence

In the newly formed sequence, each term is increased by 5. This means that the new sequence will have its first term \(a_1 + 5\) and the common difference will remain the same 'd.' The sum of the first n terms of this new sequence \(S_n'\) will then need to be determined.

Step 2 - Applying the Formula

Now apply the formula for the sum of the first \(n\) terms of an arithmetic sequence to the new sequence. The formula remains the same, i.e., \(S_n' = n/2 * (2*(a_1+5) + (n-1)*d)\). Simplify the expression to find the new sum.

Step 3 - Simplifying the Result

On further simplifying the expression, we get \(S_n' = n/2 * (2*a_1 + 2*5 + (n-1)*d) = n/2 * (2*a_1 + (n-1)*d) + 5n\). This implies that the new sum is basically the original sum plus 5 times the number of terms, since each term increased by 5.

Key Concepts

Arithmetic Progression

An arithmetic progression, also called an arithmetic sequence, is a list of numbers where each term is obtained by adding a constant value to the previous term, except for the first term, which is arbitrary. This constant value is known as the common difference, denoted by the symbol \( d \). For instance, in the sequence 2, 5, 8, 11, ..., the common difference is 3 because that's what you add to each term to get the next one.

The formula to find the \( n^{th} \) term of an arithmetic progression is given by:
\[ a_n = a_1 + (n - 1)d \]
where \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference. It's essential for students to understand how to identify the first term and the common difference in a sequence to be able to work with arithmetic progressions effectively.

Sum of an Arithmetic Series

The sum of an arithmetic series involves adding up the terms of an arithmetic sequence. To find the sum \( S_n \) of the first \( n \) terms of an arithmetic sequence, one can use the formula:
\[ S_n = \frac{n}{2}(2a_1 + (n - 1)d) \]
Here, \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the number of terms. A conceptual understanding of this formula reveals that it's akin to averaging the first and last term (\( a_1 + a_n \)) and then multiplying by the number of terms. This formula helps students quickly find the sum without having to manually add each individual term.

Sequence and Series

Sequences and series are fundamental concepts in mathematics. A sequence is a set of numbers in a specific order, while a series is the sum of a sequence of numbers. There are many types of sequences and series, but one of the simplest and most commonly studied in algebra is the arithmetic sequence and its corresponding series, the arithmetic series.

Understanding the distinction between the two is crucial. For example, the sequence 3, 7, 11, 15, ... continues indefinitely, whereas the series 3 + 7 + 11 + 15 + ... can be summed up to a finite value if we consider a specific number of terms. Efficiency in calculations can be improved by using the appropriate formulas and understanding the underlying principles of these mathematical structures.

Mathematical Induction

Mathematical induction is a powerful proof technique used to establish the truth of an infinite number of statements. In terms of sequences and series, induction can be applied to confirm the validity of formulas over all natural numbers. The process involves two primary steps: proving the base case, often when \( n = 1 \), and showing that if any statement holds for \( n \), then it also holds for \( n + 1 \), known as the inductive step.

For instance, one might use induction to prove that the formula for the sum of an arithmetic series is valid for all positive integers \( n \). This would entail verifying that the formula holds true when \( n = 1 \) and then assuming it's true for a certain integer \( k \) before proving it for \( k + 1 \). Employing mathematical induction ensures that our formula for the sum of an arithmetic series is broadly applicable and not just a coincidental pattern.