Alan

This just means add up the individual terms (and divide the result by n).  For example 

$$If the $X$'s were $X_1,X_2,X_3$ then $\sum_{k=1}^3X_k = X_1+X_2+X_3$\\\\
Sometimes you might have a function to sum, rather than a set of data; and sometimes that summation might simplify. For example\\
$$\sum_{k=1}^nk^2=\frac{n(n+1)(2n+1)}{6}$$

 I should perhaps expand that last one a little:

$$$$\sum_{k=1}^nk^2=1^2+2^2+3^2+...+n^2=\frac{n(n+1)(2n+1)}{6}$$

It occurs to me that if Jack has three children aged 1, 5 and 91 he is a remarkable person!  I'll have what he's having!!

Let n be the number of 10 paisa coins and m the number of 50 paisa coins.

We are told 

n + m = 70    ...(1)

10*n + 50*m = 19*100   ...(2) (if, that is, the r stands for rupees and there are 100 paisa in 1 rupee).

Rearrange equation (1) to get 

n = 70 - m  ...(3)

Substitute this into equation (2):

10*(70 - m) + 50*m = 19*100

Multiply out the bracketed term

700 - 10*m +50*m = 19*100

Collect the m's

700 +40*m = 19*100

Subtract 700 from both sides

40*m = 1200

Divide both sides by 40

m = 30 

Substitute this back into equation (3)

n  = 70 - 30

n = 40

Well, I suppose you could have one child who is 1; in which case you could have 1, 13 and 35; 1, 5 and 91; and 1, 7 and 65.

The other numbers that appear above are various combinations of the prime factors 5, 7 and 13 (e.g. 35 is 5*7; 91 is 7*13).

The area of a circle can be written as $$Area=\pi\frac{D^2}{4}$$ where D is the diameter.

$$$528=\pi\frac{D^2}{4}$\\
Multiply both sides by $\frac{4}{\pi}$\\
$$D^2=\frac{4*528}{\pi}$$\\
take the square root of both sides\\
$$ D=\sqrt{\frac{4*528}{\pi}}$$$$

$${\mathtt{D}} = {\sqrt{{\frac{{\mathtt{4}}{\mathtt{\,\times\,}}{\mathtt{528}}}{{\mathtt{\pi}}}}}} \Rightarrow {\mathtt{D}} = {\mathtt{25.928\: \!179\: \!257\: \!714\: \!298}}$$  feet

or D ≈ 25.93feet

 Incidentally, the area would be in square feet, not just feet!

Look for the factors of 455

$${factorize}{\left({\mathtt{455}}\right)} \Rightarrow \left\{ \begin{array}{l}{\mathtt{5}}\\
{\mathtt{7}}\\
{\mathtt{13}}\\
\end{array} \right\}$$

5*7*13 = 455

So it looks like their ages are 5, 7 and 13.

The carpenter measured the direct distance between the two marks he'd made.  If the distance was 5 inches then the corner is a right-angle.  This is because a right-angled triangle with the two legs having lengths 3 and 4 has a hypotenuse of 5. [5 = √(3² + 4²)].  

Weight is a force. Force is mass*acceleration, so: weight = 5kg*5.7m/s² = 28.5N 

Probability that nbr1 is a boy is p1 = 6/10.

Probability that nbr2 is a boy, given 5 boys left is 5/9

Probability that nbr3 is a boy given 4 boys left is 4/8

Probability that nbr 4 is a boy given 3 boys left is 3/7

$${\mathtt{Overallprobability}} = {\frac{{\mathtt{6}}{\mathtt{\,\times\,}}{\mathtt{5}}{\mathtt{\,\times\,}}{\mathtt{4}}{\mathtt{\,\times\,}}{\mathtt{3}}}{\left({\mathtt{10}}{\mathtt{\,\times\,}}{\mathtt{9}}{\mathtt{\,\times\,}}{\mathtt{8}}{\mathtt{\,\times\,}}{\mathtt{7}}\right)}} \Rightarrow {\mathtt{Overallprobability}} = {\mathtt{0.071\: \!428\: \!571\: \!428\: \!571\: \!4}}$$

Approximately 0.071 or 7.1% 

We can see from the graph below that there is no real number solution here.

6x + 8/x -3 has no values in the range -16.856 (the value at x = -2√3/3) to 10.856 (the value at x = 2√3/3)