Confusion in fraction notation $$a_n = n\dfrac{n^2 + 5}{4}$$
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):


*

*$3,8,15,24,35$

*$\dfrac{1}{2},\dfrac{2}{3},\dfrac{3}{4},\dfrac{4}{5},\dfrac{5}{6}$

*$2, 4, 8, 16 \text{ and } 32$

*$-\dfrac{1}{6},\dfrac{1}{6},\dfrac{1}{2},\dfrac{5}{6},\dfrac{7}{6}$

*$25,-125,625,-3125,15625$

*$\dfrac{3}{2},\dfrac{9}{2},\dfrac{21}{2},21,\dfrac{75}{2}$

*$65, 93$

*$\dfrac{49}{128}$

*$729$

*$\dfrac{360}{23}$

*$3, 11, 35, 107, 323$; $3+11+35+107+323+...$

*$-1,\dfrac{-1}{2},\dfrac{-1}{6},\dfrac{-1}{24},\dfrac{-1}{120}$; $-1+(\dfrac{-1}{2})+(\dfrac{-1}{6})+(\dfrac{-1}{24})+(d\frac{-1}{120})+...$

*$2, 2, 1, 0, -1$; $2+2+1+0+(-1)+...$

*$1,2,\dfrac{3}{5},\dfrac{8}{5}$

 A: 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
$$
a\frac{c}{d}:=a\cdot\frac{c}{d}
$$
A: 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}$.
A: 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 $\frac{xy}{z}$.
A: 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.
A: 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 $3+\frac14$.
A: $3 \times \frac{3^2+5}{4} = 3 \times \frac{9+5}{4} = 3\times \frac{14}{4} = 3\times \frac{7}{2} = \frac{21}{2}$
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)
A: 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.  
