MHB What does this sum symbol mean?

[shamieh]

$\sum_{i=1}^3 2 i = 12$

 

[Dethrone]

Hi shamieh,

$$\sum_{i=1}^{n=3} 2 i = 2\sum_{i}^{n=3}i=2\cdot \frac{n(n+1)}{2}=2\cdot \frac{3\cdot 4}{2}=12$$

 

[soroban]

Hello, shamieh!

Exactly what is the question?

$\displaystyle \sum_{i=1}^3 2 i = 12$

$\displaystyle \sum_{i=1}^3 2i \;=\;2(1) + 2(2) + 2(3) \;=\;2 + 4 + 6 \;=\;12$

Yes, it's true . . .

 

[MarkFL]

Out of curiosity, what is the thread title supposed to mean?

 

[phymat]

$\sum_{i=1}^3 2 i = 12$

Someone might have told the thread-starter that the symbol $$\sum$$ means summation. But, he/she might have not understood what it is. Assuming my assumption to be correct, I am telling the thread-starter what $$\sum$$ means.

For example, if you are given $$\sum_{x=0}^{5}3x+5$$, here 0 is called the lower limit and 5 is called the upper limit. It means that the expression 3x+5 will be added to itself 6 times changing the value of x every time starting from the lower limit and ending with the upper limit.
So,
$$\begin{array}{ccl}
\displaystyle\sum_{x=0}^{5}3x+5 & = & [3(0)+5]+[3(1)+5]+[3(2)+5]+[3(3)+5]+[3(4)+5]+[3(5)+5] \\
& = & 5+8+11+14+17+20 \\
& = & 75
\end{array}$$

Coming back to the original question (which is easier) :

$$\displaystyle\sum_{i=1}^3 2 i = 12$$ means that $$2i$$ will be added to itself 3 times starting from the lower limit (1) and ending with the upper limit (3).

So,
$$\displaystyle\begin{array}{ccl}\displaystyle\sum_{i=1}^3 2 i & = & 2(1)+2(2)+2(3) \\
& = & 2+4+6 \\
& = & 12
\end{array}$$