🎉 Announcing Numerade's $26M Series A, led by IDG Capital!Read how Numerade will revolutionize STEM Learning # Precalculus 7th ## David Cohen, Theodore B. Lee, David Sklar ## Chapter 14 ## Additional Topics in Algebra ## Educators ### Problem 1 Use the principle of mathematical induction to show that the statements are true for all natural numbers. $$1+2+3+\cdots+n=n(n+1) / 2$$ Check back soon! ### Problem 2 Use the principle of mathematical induction to show that the statements are true for all natural numbers. $$2+4+6+\cdots+2 n=n(n+1)$$ Check back soon! ### Problem 3 Use the principle of mathematical induction to show that the statements are true for all natural numbers. $$1+4+7+\cdots+(3 n-2)=n(3 n-1) / 2$$ Check back soon! ### Problem 4 Use the principle of mathematical induction to show that the statements are true for all natural numbers. $$5+9+13+\dots+(4 n+1)=n(2 n+3)$$ Check back soon! ### Problem 5 Use the principle of mathematical induction to show that the statements are true for all natural numbers. $$1^{2}+2^{2}+3^{2}+\cdots+n^{2}=n(n+1)(2 n+1) / 6$$ Check back soon! ### Problem 6 Use the principle of mathematical induction to show that the statements are true for all natural numbers. $$2^{2}+4^{2}+6^{2}+\cdots+(2 n)^{2}=2 n(n+1)(2 n+1) / 3$$ Check back soon! ### Problem 7 Use the principle of mathematical induction to show that the statements are true for all natural numbers. $$1^{2}+3^{2}+5^{2}+\cdots+(2 n-1)^{2}=n(2 n-1)(2 n+1) / 3$$ Check back soon! ### Problem 8 Use the principle of mathematical induction to show that the statements are true for all natural numbers. $$2+2^{2}+2^{3}+\cdots+2^{n}=2^{n+1}-2$$ Check back soon! ### Problem 9 Use the principle of mathematical induction to show that the statements are true for all natural numbers. $$3+3^{2}+3^{3}+\cdots+3^{n}=\left(3^{n+1}-3\right) / 2$$ Check back soon! ### Problem 10 Use the principle of mathematical induction to show that the statements are true for all natural numbers. $$e^{x}+e^{2 x}+e^{3 x}+\cdots+e^{n x}=\frac{e^{(n+1) x}-e^{x}}{e^{x}-1} \quad(x \neq 0)$$ Check back soon! ### Problem 11 Use the principle of mathematical induction to show that the statements are true for all natural numbers. $$1^{3}+2^{3}+3^{3}+\cdots+n^{3}=[n(n+1) / 2]^{2}$$ Check back soon! ### Problem 12 Use the principle of mathematical induction to show that the statements are true for all natural numbers. $$2^{3}+4^{3}+6^{3}+\dots+(2 n)^{3}=2 n^{2}(n+1)^{2}$$ Check back soon! ### Problem 13 Use the principle of mathematical induction to show that the statements are true for all natural numbers. $$1^{3}+3^{3}+5^{3}+\cdots+(2 n-1)^{3}=n^{2}\left(2 n^{2}-1\right)$$ Check back soon! ### Problem 14 Use the principle of mathematical induction to show that the statements are true for all natural numbers. $$\begin{array}{r} 1 \cdot 2+3 \cdot 4+5 \cdot 6+\cdots+(2 n-1)(2 n) \\=n(n+1)(4 n-1) / 3\end{array}$$ Check back soon! ### Problem 15 Use the principle of mathematical induction to show that the statements are true for all natural numbers. $$\begin{array}{r}1 \cdot 3+3 \cdot 5+5 \cdot 7+\cdots+(2 n-1)(2 n+1) \\=n\left(4 n^{2}+6 n-1\right) / 3\end{array}$$ Check back soon! ### Problem 16 Use the principle of mathematical induction to show that the statements are true for all natural numbers. $$\begin{array}{r} \frac{1}{1 \times 3}+\frac{1}{2 \times 4}+\frac{1}{3 \times 5}+\dots+\frac{1}{n(n+2)} \\=\frac{n(3 n+5)}{4(n+1(n+2)}\end{array}$$ Check back soon! ### Problem 17 Use the principle of mathematical induction to show that the statements are true for all natural numbers. $$1+\frac{3}{2}+\frac{5}{2^{2}}+\frac{7}{2^{3}}+\dots+\frac{2 n-1}{2^{n-1}}=6-\frac{2 n+3}{2^{n-1}}$$ Check back soon! ### Problem 18 Use the principle of mathematical induction to show that the statements are true for all natural numbers. $$1+2 \cdot 2+3 \cdot 2^{2}+4 \cdot 2^{3}+\cdots+n \cdot 2^{n-1}=(n-1) 2^{n}+1$$ Check back soon! ### Problem 19 Show that$n \leq 2^{n-1}$for all natural numbers$n$Check back soon! ### Problem 20 Show that 3 is a factor of$n^{3}+2 n$for all natural numbers$n$Check back soon! ### Problem 21 Show that$n^{2}+4<(n+1)^{2}$for all natural numbers$n \geq 2$Check back soon! ### Problem 22 Show that$n^{3}>(n+1)^{2}$for all natural numbers$n \geq 3$Check back soon! ### Problem 23 Prove that the statement is true for all natural numbers in the specified range. Use a calculator to carry out Step 1 $$(1.5)^{n}>2 \pi, n \geq 7$$ Check back soon! ### Problem 24 Prove that the statement is true for all natural numbers in the specified range. Use a calculator to carry out Step 1 $$(1.25)^{n}>n, n \geq 11$$ Check back soon! ### Problem 25 Prove that the statement is true for all natural numbers in the specified range. Use a calculator to carry out Step 1 $$(1.1)^{n}>n, n \geq 39$$ Check back soon! ### Problem 26 Prove that the statement is true for all natural numbers in the specified range. Use a calculator to carry out Step 1 $$(1.1)^{n}>5 n, n \geq 60$$ Check back soon! ### Problem 27 $$\text { Let } f(n)=\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\cdots+\frac{1}{n(n+1)}$$ (a) Complete the following table. \begin{tabular}{llllll} \hline$n$& 1 & 2 & 3 & 4 & 5 \\$f(n)$& & & & & \\\hline\end{tabular} (b) On the basis of the results in the table, what would you guess to be the value of$f(6) ?$Compute$f(6)$to see whether this is correct. (c) Make a conjecture about the value of$f(n),$and prove it using mathematical induction. Check back soon! ### Problem 28 Let$f(n)=\frac{1}{1 \cdot 3}+\frac{1}{3 \cdot 5}+\frac{1}{5 \cdot 7}+\cdots+\frac{1}{(2 n-1)(2 n+1)}$(a) Complete the following table. $$\begin{array}{lcccc}\hline n & 1 & 2 & 3 & 4 \\f(n) & & & & \\\hline\end{array}$$ (b) On the basis of the results in the table, what would you guess to be the value of$f(5) ?$Compute$f(5)$to see whether your guess is correct. (c) Make a conjecture about the value of$f(n),$and prove it using mathematical induction. Check back soon! ### Problem 29 Suppose that a function$f$satisfies the following conditions: $$\begin{array}{l} (1)=1 \\f(n)=f(n-1)+2 \sqrt{f(n-1)}+1 \quad(n \geq 2)\end{array}$$ (a) Complete the table. $$\begin{array}{llllll}\hline n & 1 & 2 & 3 & 4 & 5 \\f(n) & & & & & \\\hline\end{array}$$ (b) On the basis of the results in the table, what would you guess to be the value of$f(6) ?$Compute$f(6)$to see whether your guess is correct. (c) Make a conjecture about the value of$f(n)$when$n$is a natural number, and prove the conjecture using mathematical induction. Check back soon! ### Problem 30 This exercise demonstrates the necessity of carrying out both Step 1 and Step 2 before considering an induction proof valid. (a) Let$P_{n}$denote the statement that$n^{2}+1$is even. Check that$P_{1}$is true. Then give an example showing that$P_{n}$is not true for all$n$(b) Let$Q_{n}$denote the statement that$n^{2}+n$is odd. Show that Step 2 of an induction proof can be completed in this case, but not Step 1 Check back soon! ### Problem 31 A prime number is a natural number that has no factors other than itself and 1. For technical reasons, 1 is not considered a prime. Thus, the list of the first seven primes looks like this:$2,3,5,7,11,13,17 .$Let$P_{n}$be the statement that$n^{2}+n+11$is prime. Check that$P_{n}$is true for all values of$n$less than$10 .$Check that$P_{10}$is false. Check back soon! ### Problem 32 Prove that if$x \neq 1$$$1+2 x+3 x^{2}+\cdots+n x^{n-1}=\frac{1-x^{n}}{(1-x)^{2}}-\frac{n x^{n}}{1-x}$$ for all natural numbers$n$Check back soon! ### Problem 33 If$r \neq 1,$show that $$1+r+r^{2}+\cdots+r^{n-1}=\frac{r^{n}-1}{r-1}$$ for all natural numbers$n$Check back soon! ### Problem 34 Use mathematical induction to show that $$x^{n}-1=(x-1)\left(1+x+x^{2}+\cdots+x^{n-1}\right)$$ for all natural numbers$n$Check back soon! ### Problem 35 Prove that 5 is a factor of$n^{5}-n$for all natural numbers$n \geq 2$Check back soon! ### Problem 36 Prove that 4 is a factor of$5^{n}+3$for all natural numbers$n .$Check back soon! ### Problem 37 Prove that 5 is a factor of$2^{2 n+1}+3^{2 n+1}$for all nonnegative integers$n$Check back soon! ### Problem 38 Prove that 8 is a factor of$3^{2 n}-1$for all natural numbers$n .$Check back soon! ### Problem 39 Prove that 3 is a factor of$2^{n+1}+(-1)^{n}$for all nonnegative integers$n$Check back soon! ### Problem 40 Prove that 6 is a factor of$n^{3}+3 n^{2}+2 n$for all natural numbers$n$Check back soon! ### Problem 41 Use mathematical induction to show that$x-y$is a factor of$x^{n}-y^{n}$for all natural numbers$n .$Suggestion for Step 2: Verify and then use the fact that $$x^{k+1}-y^{k+1}=x^{k}(x-y)+\left(x^{k}-y^{k}\right) y$$ Check back soon! ### Problem 42 Use mathematical induction to prove that the formulas hold for all natural numbers$n$. $$\log _{10}\left(a_{1} a_{2} \ldots a_{n}\right)=\log _{10} a_{1}+\log _{10} a_{2}+\dots+\log _{10} a_{n}$$ Check back soon! ### Problem 43 Use mathematical induction to prove that the formulas hold for all natural numbers$n\$.
$$(1+p)^{n} \geq 1+n p, \text { where } p>-1$$

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