Problem 33 - Algebra 2

Question

Prove that \(4+7+10+\cdots+(3 n+1)=\frac{n(3 n+5)}{2}\) for all positive integers \(n\).

Step-by-Step Solution

Answer

The series sum is \(\frac{n(3n + 5)}{2}\), which matches the given equation, proving it true.

Step 1: Understanding the Sequence

The sequence given is an arithmetic sequence where each term can be expressed as \(3n + 1\), for successive integers. The first term, \(a\), is 4 (when \(n = 1\)), and the common difference, \(d\), is 3.

Step 2: Formulating the Series Sum

The problem states the sum to be proven as \(S = 4 + 7 + 10 + \cdots + (3n + 1)\). We aim to prove that this is equal to \(\frac{n(3n + 5)}{2}\). An arithmetic series sum can be expressed using the formula: \(S_n = \frac{n}{2} (2a + (n-1)d)\).

Step 3: Apply the Arithmetic Sequence Sum Formula

Substitute the values into the arithmetic series formula. We have \(a = 4\), \(d = 3\), and the \(n\)-th term is \(3n + 1\). The number of terms \(n\) equals \(( \frac{3n + 1 - 4}{3}) + 1 = n\).

Step 4: Simplify the Sum Formula with Substituted Values

Directly using the formula for an arithmetic sequence sum: \[S_n = \frac{n}{2} (2a + (n-1)d),\]substitute \(a = 4\) and \(d = 3\) to get:\[S_n = \frac{n}{2} (2\cdot4 + (n-1)\cdot3).\]

Step 5: Simplifying the Expression

Continue simplifying:\[S_n = \frac{n}{2} (8 + 3n - 3) = \frac{n}{2} (3n + 5).\]We see that this matches the form given in the problem statement, completing the proof.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a list of numbers where each term after the first is obtained by adding a fixed, constant number called the "common difference" to the previous term. In our initial arithmetic sequence, the terms are shown as \(4, 7, 10, \ldots, (3n + 1)\). Here, the sequence is derived from the expression \(3n + 1\), where \(n\) is a positive integer.
The first term of the sequence, denoted by \(a\), is \(4\), corresponding to the starting position with \(n = 1\).
As you notice the difference between successive terms, \( \text{such as} \ 7 - 4 \ or \ 10 - 7 \), it remains constant at \(3\). This constant rate of increase or decrease is what forms the foundation of an arithmetic sequence.

Sequence Sum Formula

The sum of an arithmetic sequence, also known as an arithmetic series, can be found with a specific formula. This formula helps to efficiently calculate the sum of all the terms without individually adding each one.
The general formula for the sum of the first \(n\) terms of an arithmetic series is given by:
\[S_n = \frac{n}{2} (2a + (n-1)d),\]
where:

• \(S_n\) is the sum of the first \(n\) terms,
• \(a\) is the first term of the sequence,
• \(d\) is the common difference between the successive terms, and
• \(n\) is the number of terms to be added.

The formula effectively pairs the terms from the start and end towards the middle, decreasing the repetition and simplifying the addition process.

Common Difference

The common difference in an arithmetic sequence is the consistent interval between consecutive terms. In our initial sequence problem, each term increases by a value of \(3\). This constant value is what we refer to as the common difference and is symbolized by \(d\).
Understanding the common difference is crucial because it dictates the nature of the sequence. Whether the terms are increasing or decreasing, the common difference follows as a constant value.
It is also used in the sequence sum formula, allowing us to generalize the calculation of sums for any arithmetic series with known parameters.

Proof by Induction

Proof by induction is a powerful mathematical tool used to prove statements valid for all positive integers. Inductive proofs consist of two key steps: the base case and the inductive step.
For our series sum problem, to use induction, one would:

• Start by proving the base case where \(n = 1\), showing that the equation holds.
• Assume it is true for \(n = k\); this is the inductive hypothesis.
• Then prove it for \(n = k + 1\), completing the inductive step.

Successfully completing these steps solidifies that the sum formula is valid for all \(n\), confirming the closed-form expression of the series sum. Induction assures a logical flow to verifying problems in mathematics, encapsulating the transition from known terms to those unknown.