Question

If
$\frac{9 {}^{n - 1} \times {3}^{ \frac{ - n} {2} { - 2}^{} } + {27}^{n} }{ < {3}^{n} + 2 > {}^{3} } \: \: = \: \: \frac{1}{729}$
Find the value of m and n​

Answer 1

Answer:

Symbol Command

$\frac {1}{2}$ \frac{1}{2} or \frac12

$\frac{2}{x+2}$ \frac{2}{x+2}

$\frac{1+\frac{1}{x}}{3x + 2}$ \frac{1+\frac{1}{x}}{3x + 2}

Notice that with fractions with a 1-digit numerator and a 1-digit denominator, we can simply group the numerator and the denominator together as one number. However, for fractions with either a numerator or a denominator that requires more than one character (or if the numerator starts with a letter), you need to surround everything in curly brackets.

Use \cfrac for continued fractions.

Expression Command

$\cfrac{2}{1+\cfrac{2}{1+\cfrac{2}{1+\cfrac{2}{1}}}}$ \cfrac{2}{1+\cfrac{2}{1+\cfrac{2}{1+\cfrac{2}{1}}}}

Radicals

Symbol Command

$\sqrt{3}$ \sqrt{3}

$\sqrt{x+y}$ \sqrt{x+y}

$\sqrt{x+\frac{1}{2}}$ \sqrt{x+\frac{1}{2}}

$\sqrt[3]{3}$ \sqrt[3]{3}

$\sqrt[n]{x}$ \sqrt[n]{x}

Sums, Products, Limits and Logarithms

Use the commands \sum, \prod, \lim, and \log respectively. To denote lower and upper bounds, or the base of the logarithm, use _ and ^ in the same way they are used for subscripts and superscripts. (Lower and upper bounds for integrals work the same way, as you'll see in the calculus section)

Symbol Command

$\textstyle \sum_{i=1}^{\infty}\frac{1}{i}$ \sum_{i=1}^{\infty}\frac{1}{i}

$\textstyle \prod_{n=1}^5\frac{n}{n-1}$ \prod_{n=1}^5\frac{n}{n-1}

$\textstyle \lim_{x\to\infty}\frac{1}{x}$ \lim_{x\to\infty}\frac{1}{x}

$\textstyle \lim\limits_{x\to\infty}\frac{1}{x}$ \lim\limits_{x\to\infty}\frac{1}{x}

$\textstyle \log_n n^2$ \log_n n^2

Some of these are prettier in display mode:

Symbol Command

$\sum_{i=1}^{\infty}\frac{1}{i}$ \sum_{i=1}^{\infty}\frac{1}{i}

$\prod_{n=1}^5\frac{n}{n-1}$ \prod_{n=1}^5\frac{n}{n-1}

$\lim_{x\to\infty}\frac{1}{x}$ \lim_{x\to\infty}\frac{1}{x}

Note that we can use sums, products, and logarithms without _ or ^ modifiers.

Symbol Command

$\sum\frac{1}{i}$ \sum\frac{1}{i}

$\frac{n}{n-1}$ \frac{n}{n-1}

$\textstyle \log n^2$ \log n^2

$\textstyle \ln e$ \ln e

Mods

Answer 2

$\LARGE\underline{\blue{\sf{Question:-}}}$

$\LARGE\underline{\pink{\sf{Answer:-}}}$

Answer:

Symbol Command

$\frac {1}{2}$ \frac{1}{2} or \frac12

$\frac{2}{x+2}$ \frac{2}{x+2}

$\frac{1+\frac{1}{x}}{3x + 2}$ \frac{1+\frac{1}{x}}{3x + 2}

Notice that with fractions with a 1-digit numerator and a 1-digit denominator, we can simply group the numerator and the denominator together as one number. However, for fractions with either a numerator or a denominator that requires more than one character (or if the numerator starts with a letter), you need to surround everything in curly brackets.

Use \cfrac for continued fractions.

Expression Command

$\cfrac{2}{1+\cfrac{2}{1+\cfrac{2}{1+\cfrac{2}{1}}}}$ \cfrac{2}{1+\cfrac{2}{1+\cfrac{2}{1+\cfrac{2}{1}}}}

Radicals

Symbol Command

$\sqrt{3}$ \sqrt{3}

$\sqrt{x+y}$ \sqrt{x+y}

$\sqrt{x+\frac{1}{2}}$ \sqrt{x+\frac{1}{2}}

$\sqrt[3]{3}$ \sqrt[3]{3}

$\sqrt[n]{x}$ \sqrt[n]{x}

Sums, Products, Limits and Logarithms

Use the commands \sum, \prod, \lim, and \log respectively. To denote lower and upper bounds, or the base of the logarithm, use _ and ^ in the same way they are used for subscripts and superscripts. (Lower and upper bounds for integrals work the same way, as you'll see in the calculus section)

Symbol Command

$\textstyle \sum_{i=1}^{\infty}\frac{1}{i}$ \sum_{i=1}^{\infty}\frac{1}{i}

$\textstyle \prod_{n=1}^5\frac{n}{n-1}$ \prod_{n=1}^5\frac{n}{n-1}

$\textstyle \lim_{x\to\infty}\frac{1}{x}$ \lim_{x\to\infty}\frac{1}{x}

$\textstyle \lim\limits_{x\to\infty}\frac{1}{x}$ \lim\limits_{x\to\infty}\frac{1}{x}

$\textstyle \log_n n^2$ \log_n n^2

Some of these are prettier in display mode:

Symbol Command

$\sum_{i=1}^{\infty}\frac{1}{i}$ \sum_{i=1}^{\infty}\frac{1}{i}

$\prod_{n=1}^5\frac{n}{n-1}$ \prod_{n=1}^5\frac{n}{n-1}

$\lim_{x\to\infty}\frac{1}{x}$ \lim_{x\to\infty}\frac{1}{x}

Note that we can use sums, products, and logarithms without _ or ^ modifiers.

Symbol Command

$\sum\frac{1}{i}$ \sum\frac{1}{i}

$\frac{n}{n-1}$ \frac{n}{n-1}

$\textstyle \log n^2$ \log n^2

$\Answer:

Symbol Command

$\frac {1}{2}$ \frac{1}{2} or \frac12

$\frac{2}{x+2}$ \frac{2}{x+2}

$\frac{1+\frac{1}{x}}{3x + 2}$ \frac{1+\frac{1}{x}}{3x + 2}

Notice that with fractions with a 1-digit numerator and a 1-digit denominator, we can simply group the numerator and the denominator together as one number. However, for fractions with either a numerator or a denominator that requires more than one character (or if the numerator starts with a letter), you need to surround everything in curly brackets.

Use \cfrac for continued fractions.

Expression Command

$\cfrac{2}{1+\cfrac{2}{1+\cfrac{2}{1+\cfrac{2}{1}}}}$ \cfrac{2}{1+\cfrac{2}{1+\cfrac{2}{1+\cfrac{2}{1}}}}

Radicals

Symbol Command

$\sqrt{3}$ \sqrt{3}

$\sqrt{x+y}$ \sqrt{x+y}

$\sqrt{x+\frac{1}{2}}$ \sqrt{x+\frac{1}{2}}

$\sqrt[3]{3}$ \sqrt[3]{3}

$\sqrt[n]{x}$ \sqrt[n]{x}

Sums, Products, Limits and Logarithms

Use the commands \sum, \prod, \lim, and \log respectively. To denote lower and upper bounds, or the base of the logarithm, use _ and ^ in the same way they are used for subscripts and superscripts. (Lower and upper bounds for integrals work the same way, as you'll see in the calculus section)

Symbol Command

$\textstyle \sum_{i=1}^{\infty}\frac{1}{i}$ \sum_{i=1}^{\infty}\frac{1}{i}

$\textstyle \prod_{n=1}^5\frac{n}{n-1}$ \prod_{n=1}^5\frac{n}{n-1}

$\textstyle \lim_{x\to\infty}\frac{1}{x}$ \lim_{x\to\infty}\frac{1}{x}

$\textstyle \lim\limits_{x\to\infty}\frac{1}{x}$ \lim\limits_{x\to\infty}\frac{1}{x}

$\textstyle \log_n n^2$ \log_n n^2

Some of these are prettier in display mode:

Symbol Command

$\sum_{i=1}^{\infty}\frac{1}{i}$ \sum_{i=1}^{\infty}\frac{1}{i}

$\prod_{n=1}^5\frac{n}{n-1}$ \prod_{n=1}^5\frac{n}{n-1}

$\lim_{x\to\infty}\frac{1}{x}$ \lim_{x\to\infty}\frac{1}{x}

Note that we can use sums, products, and logarithms without _ or ^ modifiers.

Symbol Command

$\sum\frac{1}{i}$ \sum\frac{1}{i}

$\frac{n}{n-1}$ \frac{n}{n-1}

$\textstyle \log n^2$ \log n^2

$\textstyle \ln e$ \ln e

Mods e

Mods