Practice (23)

2154

Simplify

$\displaystyle\frac{2^2-2}{2^2+2}\cdot\displaystyle\frac{3^2-3}{3^2+3}\cdots\displaystyle\frac{10^2-2}{10^2+10}$

2156

Compute

$\Large(\sqrt{6+4\sqrt{2}} + \sqrt{6-4\sqrt{2}}\Large)^2$

3185

Show that

$\frac{1}{1\times 2\times 3}+\frac{1}{2\times 3\times 4}+\cdots + \frac{1}{n\times (n+1)\times (n+2)}=\frac{n(n+3)}{4(n+1)(n+2)}$

3232

Find the value of

$x^3+x^2y+xy^2+y^3$ if

$x=\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$ and

$y=\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$ .

3233

Simplify

$\sqrt{10+4\sqrt{3-2\sqrt{2}}}$

3234

Let

$\sqrt{1+\sqrt{21+12\sqrt{3}}}=\sqrt{a}+\sqrt{b}$ . Find

$a+b$ .

3235

Let

$a\ge 0, n\ge 0,$ and

$m > 0$ . Show that

$\sqrt{a+m}+\sqrt{a+m+n} > \sqrt{a} + \sqrt{a+2m+n}$ .

3236

Simplify

$(\sqrt{10}+\sqrt{11}+\sqrt{12})(\sqrt{10}+\sqrt{11}-\sqrt{12})(\sqrt{10}-\sqrt{11}+\sqrt{12})(-\sqrt{10}+\sqrt{11}+\sqrt{12})$

3237

Simplify

$(\sqrt[3]{3}+\sqrt[3]{2})(\sqrt[3]{9}-\sqrt[3]{6}+\sqrt[3]{4})$

3239

Simplify

$\sqrt{1 + 1995\sqrt{4 + 1995 \cdot 1999}}$ .

3240

Evaluate

$\sqrt{5+\sqrt{5^2+\sqrt{5^4+\sqrt{5^8+...}}}}$

3241

Simplify

$\sqrt{1-\frac{\sqrt{3}}{2}}$

3242

Simplify

$\frac{1}{2+\frac{1}{2+\cdots}}$

3243

Simplify

$(\sqrt{2})^{(\sqrt{2})^{(\sqrt{2})^{\cdots}}}$

3244

Simplify

$2^{\sqrt{2^{\sqrt{2^{\sqrt{2}^{\cdots}}}}}}$

3245

Compute

$\frac{1}{\frac{1}{\frac{1}{\cdots}+1+\frac{1}{\cdots}}+1+\frac{1}{\frac{1}{\cdots}+1+\frac{1}{\cdots}}}$

3246

Use at least two ways to prove

$\sqrt{x\sqrt{x\sqrt{x\sqrt{\cdots}}}}=x$

3247

Show that

$\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}=\frac{1}{\frac{1}{1+\frac{1}{1+\cdots}}}=\frac{1+\sqrt{5}}{2}$

3248

Show that, if both converge,

$\sqrt{a+b\sqrt{a+b\sqrt{a+\cdots}}}=b+\frac{a}{b+\frac{a}{b+\cdots}}=\frac{b+\sqrt{b^2+4a}}{2}$

3249

Compute

$\sqrt{\frac{2}{2^2}+\sqrt{\frac{2}{2^4}+\sqrt{\frac{2}{2^8}+\cdots}}}$

3250

Compute

$\sqrt{\frac{2}{2^1}+\sqrt{\frac{2}{2^2}+\sqrt{\frac{2}{2^4}+\cdots}}}$

3252

Compute

$\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots}}}}$

3253

Compute $\sqrt{6+2\sqrt{7+3\sqrt{8+\cdots}}}$

3254

Without using a calculator, explain that $\sqrt{20+\sqrt{20+\sqrt{20}}}-\sqrt{20-\sqrt{20-\sqrt{20}}}\approx 1$

3255

Show that

$\sqrt{n+\sqrt{n+\sqrt{n+\cdots}}}=\frac{1+\sqrt{1+4n}}{2}$ and

$\sqrt{n-\sqrt{n-\sqrt{n-\cdots}}}=\frac{-1+\sqrt{1+4n}}{2}$

PolynomialAndEquation InfiniteRepitition SpecialSequence

With Solutions

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