sir I am very weak in reading sum notation, like reading kanji words to me, but I tried to make some sense of it. I dont know where is 7114 coming from? why are you equating it?
I am sorry that my answer was not helpful. This is why it is so important that you tell us about your mathematical knowledge.
I was having trouble getting the math symbols to come out right last night. The symbol you did not understand should have been
[MATH]\sum_{8}^{71} 14 * j.[/MATH]
Let's start with a simple idea. I want to add up the following
[MATH](1 * 14) + (2 * 14) + (3 * 14) + (4 * 14).[/MATH]
One way is to do 4 multiplications and then do 3 additions
[MATH](1 * 14) + (2 * 14) + (3 * 14) + (4 * 14) = 14 + 28 + 42 + 56 = 140.[/MATH]
A different way is to do 3 additions and then 1 multiplication.
[MATH](1 * 14) + (2 * 14) + (3 * 14) + (4 * 14) = 14 * (1 + 2 + 3 + 4) = 14 * 10 = 140.[/MATH]
Notice that the second way has fewer computations, 4 versus 7.
There is a third way, which is to use a formula for adding up succesive integers. That way involves 2 multiplications and one division.
[MATH](1 * 14) + (2 * 14) + (3 * 14) + (4 * 14) = 14 * (1 + 2 + 3 + 4) = 14 * \dfrac{4 * 5}{2} = 14 * \dfrac{20}{2} = 14 * 10 = 140.[/MATH]
Notice that the third way has even fewer computations, 3 versus 4 versus 7.
The more products to be added up, the greater the disparity in computations among the three methods.
And your problem has a lot of products. So we use the third method, which will always require only 2 multiplications and 1 division.
So here is the general formula for adding up n integers starting with 1:
[MATH]\dfrac{n * (n + 1)}{2}.[/MATH]
Do you understand so far?
So how would I write the sum of all the multiples of 14 less than 1000? Well, I am
DEFINITELY going to use the third method, but first I need to figure out how many numbers are involved. What is the largest multiple of 14 less than 1000. Pka showed you how to determine that by using floor notation.
[MATH]\dfrac{1000}{14} = 71 + \dfrac{6}{14} \implies \left \lfloor \dfrac{1000}{14} \right \rfloor = 71.[/MATH]
Let's check.
[MATH]14 * 71 = 994 \text { and } 994 + 14 = 1008.[/MATH]
So there are 71 multiples of 14 less than 1000. So how do I compute the sum of all the multiples of 14 under 1000?
[MATH](14 + 28 + ... 994) = 14 * (1 + 2 + ... 71) = 14 * \dfrac{71 * 72}{2} = 14 * \dfrac{5112}{2} = 7 * 5112 = 35784.[/MATH]
But I do not want to include multiples of 14 less than 100 in my total. What is their sum?