Confusion in fraction notation

Question

$$a_n = n\dfrac{n^2 + 5}{4}$$

In the above fraction series, for $n=3$ I think the answer should be $26/4$, while the answer in the answer book is $21/2$ (or $42/4$). I think the difference stems from how we treat the first $n$. In my understanding, the first number is a complete part and should be added to fraction, while the book treats it as part of fraction itself, thus multiplying it with $n^2+5$. So, I just want to understand which convention is correct.

This is from problem 6 in exercise 9.1 on page 180 of the book Sequences and Series.

Here is the answer sheet from the book (answer 6, 3rd element):

1. $3,8,15,24,35$
2. $\dfrac{1}{2},\dfrac{2}{3},\dfrac{3}{4},\dfrac{4}{5},\dfrac{5}{6}$
3. $2, 4, 8, 16 \text{ and } 32$
4. $-\dfrac{1}{6},\dfrac{1}{6},\dfrac{1}{2},\dfrac{5}{6},\dfrac{7}{6}$
5. $25,-125,625,-3125,15625$
6. $\dfrac{3}{2},\dfrac{9}{2},\dfrac{21}{2},21,\dfrac{75}{2}$
7. $65, 93$
8. $\dfrac{49}{128}$
9. $729$
10. $\dfrac{360}{23}$
11. $3, 11, 35, 107, 323$; $3+11+35+107+323+...$
12. $-1,\dfrac{-1}{2},\dfrac{-1}{6},\dfrac{-1}{24},\dfrac{-1}{120}$; $-1+(\dfrac{-1}{2})+(\dfrac{-1}{6})+(\dfrac{-1}{24})+(d\frac{-1}{120})+...$
13. $2, 2, 1, 0, -1$; $2+2+1+0+(-1)+...$
14. $1,2,\dfrac{3}{5},\dfrac{8}{5}$

Answer 1

in elementary school math the fraction $x\frac{y}{z}$ usually means $x+\frac{y}{z}$ and is called a mixed fraction.

However these are almost never used after junior high.

Most of the time when you see $x\frac{y}{z}$ the two terms should be multiplied, so it is equal to $\frac{xy}{z}$.

Answer 2

$3 \times \frac{3^2+5}{4} = 3 \times \frac{9+5}{4} = 3\times \frac{14}{4} = 3\times \frac{7}{2} = \frac{21}{2}$

Normally (even though in calculator this is often not true) the convention is that a number on the side of a fraction is multiplying that fraction. There should be a "times" either a cross or a dot, however often you can omit it as a shortcut (in veritas, it is almost always omitted apart from very specific cases when someone wants to emphasise the steps as in my answer above)

Answer 3

It just depends on context.

In some rare cases $$ a\frac{c}{d}:=a+\frac{c}{d} $$ which is the interpretation in your answer, but mostly $$ a\frac{c}{d}:=a\cdot\frac{c}{d} $$

Answer 4

I don't think I've ever seen $x \frac{y}{z}$ used to mean $x + \frac{y}{z}$ except when $x$, $y$ and $z$ are literal integers (e.g. $2 \frac{3}{4}$). That's not to say it never happens, but it would be terribly confusing.

Answer 5

I don't think there can ever be a mixed fraction of the form $n\frac{n^2+5}{4}$ if $n$ $\in$ $\mathbb{N}$. Please note that if it were a mixed fraction then $n^2+5$ would denote the remainder while $4$ is the divisor and this would never be possible as for $n$ $\in$ $\mathbb{N}$, $n^2+5 \gt 4$ always.

Hence, this expression would definitely denote $n\times$$\frac{n^2+5}{4}$.

Answer 6

Here is my attempt at a helpful rule:

The expression $x\frac{y}{z}$ always means $x\times\frac{y}{z}$ except when $x,y,$ and $z$ are all integers written in decimal notation; then it means $x+\frac{y}{z}$.

So $n\frac{n^2+5}{4}$ means $n\times\frac{n^2+5}{4}$, but $3\frac14$ means $3+\frac14$.

Answer 7

Here's the Cliff Notes version. Actually cut and pasted from Cliff Notes: "•Two variables (letters) next to each other: ab means a times b"

This is always the case, even if the variable is identified as an integer. If a=3 and b=4 then ab=12, NOT 34. To understand why 3 1/4 is not 3 x 1/4 it is actually simple - it follows a different rule of notation. when a number is in front of a fraction they are considered parts of a total number that consists of a whole and a fractional part. One final example of notation. Intuitively we know that 34 is thirty-four. it's not 7, ie 3+4, and it's not 12, ie 3x4. The notation is our common base 10 system: so 34 is just an abbreviation for (3x10) + (4x1). Since we do it so often we don't think about it, it's intuitive and we don't look for exceptions. That's how you should treat ab: it's a times b always.