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Chapter 9: Problem 5

Prove that the statement is true for every positive integer \(n\). $$2+7+12+\dots+(5 n-3)=\frac{1}{2} n(5 n-1)$$

Short Answer

Expert verified
The statement is true for all positive integers by induction.

Step by step solution

01

Understand the Problem

You are given a series: \(2 + 7 + 12 + \ldots + (5n - 3)\) and you need to prove that the sum of the first \(n\) terms of this series equals \(\frac{1}{2}n(5n-1)\). This is done using mathematical induction.
02

Base Case

Check if the statement is true for \(n = 1\). Substitute \(n = 1\) into both the series and the formula.For the series: The first term is \(2\).For the formula: \(\frac{1}{2} \cdot 1 (5 \cdot 1 - 1) = \frac{1}{2} (5 - 1) = \frac{1}{2} \cdot 4 = 2\).Since both are equal, the base case holds.
03

Inductive Hypothesis

Assume the statement is true for some positive integer \(k\); that is,\[2 + 7 + 12 + \ldots + (5k - 3) = \frac{1}{2} k (5k - 1)\].This is known as the induction hypothesis.
04

Inductive Step

Show that if the statement holds for \(k\), it also holds for \(k+1\). This involves proving:\[ 2 + 7 + 12 + \ldots + (5k - 3) + (5(k+1) - 3) = \frac{1}{2} (k+1)(5(k+1) - 1) \].
05

Apply Inductive Hypothesis

Using the hypothesis, the sum\[ 2 + 7 + 12 + \ldots + (5k - 3) = \frac{1}{2} k (5k - 1)\],so we need to prove:\[\frac{1}{2} k(5k - 1) + (5(k+1) - 3) = \frac{1}{2} (k+1)(5k + 4) \].
06

Simplify the Left Side

Compute \(5(k+1) - 3 = 5k + 5 - 3 = 5k + 2\).Substitute this into the equation:\[\frac{1}{2} k (5k - 1) + 5k + 2 \].Simplify to:\[\frac{1}{2}[k(5k - 1) + 2(5k + 2)] \].
07

Simplify the Expression

Now expand and simplify:\[\frac{1}{2}[5k^2 - k + 10k + 4] = \frac{1}{2}[5k^2 + 9k + 4]\],simplifying further:\[\frac{1}{2}[5k^2 + 9k + 4]\].
08

Right Side Expression

The right side is \(\frac{1}{2} (k+1)(5k+4)\), expand and simplify:\[= \frac{1}{2}[5k^2 + 4k + 5k + 4]\].This simplifies to \[\frac{1}{2}[5k^2 + 9k + 4]\].
09

Conclusion

Both sides simplify to the same expression: \[\frac{1}{2}(5k^2 + 9k + 4)\].Hence, by induction, the statement is true for all positive integers \(n\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Series Summation
Understanding series summation is crucial when working with sequences. In this problem, we are tasked with finding the sum of a specific arithmetic series given by the formula: \(5n - 3\). Here, each term of the series increases linearly as "n" increases. For instance, the series starts as \(2, 7, 12, \ldots\) and so on.
  • The first term \(a = 2\).
  • The common difference \(d\) is the difference between consecutive terms, which is \(5\).
By understanding the pattern, we can find the sum of the series up to the nth term and eventually prove that the sum equals \(\frac{1}{2}n(5n-1)\) for any positive integer \(n\). Knowing how to sum series is an essential skill in mathematics as it allows us to find total values over ranges of numbers.
Proof Techniques
Proving mathematical statements often involves a technique called proof by induction. This is a structured process useful for proving statements that are assumed to hold true for all positive integers. The exercise follows these steps:
  • **Base Case**: Verify the statement for the initial value, usually \(n = 1\). If it holds, you move to the next step.
  • **Inductive Step**: Assume the statement holds true for an arbitrary positive integer \(k\). Then, prove it also holds for \(k + 1\).
Using these steps stabilizes the proof process and ensures the statement is true for all natural numbers beyond the base case. Induction is powerful because it reduces proving an infinite number of cases to proving just two things: the base case and the inductive step.
Inductive Hypothesis
The inductive hypothesis is a critical part of proof by induction. It involves assuming the statement is true for a particular positive integer, say \(k\). This hypothesis acts as the temporary truth which helps in proving the subsequent step for \(k + 1\).
  • In our problem, we assume: \(2 + 7 + 12 + \ldots + (5k - 3) = \frac{1}{2} k (5k - 1)\).
  • This step is key as it lays the groundwork for justifying its truth for \(k+1\).
With this assumption, you're not immediately proving for every positive integer but establishing a base to show the pattern holds. It is a logical intermediary step designed to bridge the known and the unknown in mathematical induction.
Algebraic Manipulation
Algebraic manipulation is the process of rearranging and simplifying algebraic expressions. It is key to transforming the inductive hypothesis into a proof that includes the next term in the series. To achieve this:
  • First, you calculate the next term in the series, which is \(5(k+1) - 3\).
  • Add this new term to both sides of the inductive hypothesis to develop the equation for \(k + 1\).
  • Simplify the resulting expressions to see if both sides match.
Through algebraic manipulation, the complex expressions are broken down into recognizable forms, leading to the ultimate expression that validates the original series summation. Mastering this skill is essential in mathematical proofs and problem-solving.

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