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Chapter 1: Problem 4

Prove that \(1^{2}+3^{2}+\cdots+(2 n-1)^{2}=\left(4 n^{3}-n\right) / 3\) for all \(n \in \mathbb{N}\).

Short Answer

Expert verified
The proposition \(1^{2}+3^{2}+\cdots+(2 n-1)^{2}=\frac{4 n^{3}-n}{3}\) is shown to be true for all \(n \in \mathbb{N}\) by using mathematical induction.

Step by step solution

01

Establish the Base Case

First, we establish the base case of \(n=1\). For \(n=1\), the left-hand side (LHS) of the equation equals \(1^{2}\), which is equal to 1. The right-hand side (RHS) when \(n=1\) is \(\frac{4 \cdot 1^{3}-1} {3} = 1\). So, as LHS = RHS, the equation holds true for \(n=1\).
02

Assume true for \(n=k\)

Next, we assume that the equation holds true for \(n=k\). That is, we assume that \(1^{2}+3^{2}+\cdots+(2 k-1)^{2}=\frac{4 k^{3}-k} {3}\). This is known as the induction hypothesis.
03

Prove true for \(n=k+1\)

Now, we need to prove that the equation is true for \(n=k+1\), if it is assumed true for \(n=k\). That is, we should show (under the assumption that the statement holds for \(n=k\)), that \(1^{2}+3^{2}+\cdots+(2 k-1)^{2} + (2k+1)^2 = \frac{4 (k+1)^{3}-(k+1)} {3}\). This can be done by adding \((2k+1)^2\) to both sides of the induction hypothesis and simplifying both sides until RHS = LHS.
04

Simplify RHS

First, we simplify the RHS to form \(\frac{4(k^3+3k^2+3k+1)-(k+1)}{3} = \frac{4k^3+12k^2+12k+4-k-1}{3} = \frac{4k^3-k +12k^2}{3}+ 4k + 1\).
05

Simplify LHS

Then, we add \((2k+1)^2\) to both sides of the induction hypothesis and simplify the LHS: adding \((2k+1)^2\) to the LHS of the induction hypothesis gives \(1^{2}+3^{2}+\cdots+(2 k-1)^{2} + (2k+1)^2 = \frac{4 k^{3}-k} {3} + (2k+1)^2 = \frac{4k^3-k +12k^2}{3}+ 4k + 1\).
06

Compare LHS and RHS

The LHS and RHS are both equal to \(\frac{4k^3-k +12k^2}{3}+ 4k + 1\). This proves that if the equation holds true for \(n=k\), then it will also hold true for \(n=k+1\). Hence, by mathematical induction, the proposition holds for all natural numbers \(n\).

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Most popular questions from this chapter

Let \(f, g\) be functions such that \((g \circ f)(x)=x\) for all \(x \in D(f)\) and \((f \circ g)(y)=y\) for all \(y \in D(g) .\) Prove that \(g=f^{-1}\).

Conjecture a formula for the sum of the first \(n\) odd natural numbers \(1+3+\cdots+(2 n-1)\), and prove your formula by using Mathematical Induction.

Prove that \(3+11+\cdots+(8 n-5)=4 n^{2}-n\) for all \(n \in \mathbb{N}\).

Prove that \(1 / \sqrt{1}+1 / \sqrt{2}+\cdots+1 / \sqrt{n}>\sqrt{n}\) for all \(n \in \mathbb{N}\).

Prove that a nonempty set \(T_{1}\) is finite if and only if there is a bijection from \(T_{1}\) onto a finite set \(T_{2}\)
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