Sn the sum of the first n odd numbers

Question

i need help to solve a question that say :

Sn the sum of the first n odd numbers

1 - Formalize this sum: (Give mathematical expression)?

2 - Calculate this sum?

how to solve that ? :/

Answer 1

HINT

• consider the sum of the first $2n$ numbers, that is

$$\sum_{k=1}^{2n} k=1+2+3+\ldots+(2n-1)+2n$$

• subtract the sum of the first $n$ even numbers picking out a factor 2

$$2+4+6+\ldots+(2n-2)+2n=2\sum_{k=1}^{n} k$$

Answer 2

There is a very neat pictorial proof of the solution dated as far back as the Pythagorean schools who considered every number to have a shape apparently.

Try drawing the odd numbers as "L" shapes of dots,

i.e. \begin{array}{cc} \blacksquare \\ \\ \end{array} \begin{array}{cc} \blacksquare \\ \blacksquare & \blacksquare\\ \end{array} \begin{array}{cc} \blacksquare \\ \blacksquare \\ \blacksquare & \blacksquare & \blacksquare\\ \end{array}for 1, 3, and 5 and so on, then see what happens when you put them together.

p.s. apologies for my LaTeX, if someone can make it look prettier by all means go ahead and edit. Thanks

Answer 3

Here's the $A=B$ solution: $S_1 = 1 = 1 = 1^2$, $S_2 = 1 + 3 = 4 = 2^2$, $S_3 = 1 + 3 + 5 = 9 = 3^2$. Taking differences, we see that the formula must be a quadratic in $n$; three values of a quadratic determine the quadratic: we must have $S_n = n^2$ for all $n$.