00:02
Now let's look at an example where we are asked to find the summation i equal 1 to 25 of i cubed minus 2i, and we are to use the properties of summation instead of just plugging in the values for i and expanding the whole thing out.
00:19
Now, some of the properties that we're going to use particularly for this question is because there are two terms in the parentheses, we have the property that the summation i equal 1 to end of a sub i either add or subtract which is the case you have b sub i when i have a summation of several terms it's the summation of each of the terms individually so it's a summation i equal one to n of a sub i and then if it's add you'll add if it's subtract you'll subtract the summation i equal 1 to n of b sub i next up we also have that the summation i equal 1 to n of a constant k times a sub i well constant factors in every single term would be a common factor that you could factor out of the expression and that ends up being able to factor the k in front of the summation and we get k times the summation i equal one to n of a sub i so i can factor out common factors that are constants and then the summation i equal one to n of a number is the number times the tap out value so that's our c times our n this one in particular we won't need for this specific question, but it's always good to review properties that we use quite frequently.
02:02
We have the summation i equal 1 to n of i is equal to n times n plus 1 over 2.
02:11
The summation i equal 1 to n of i square is n times n plus 1 times 2 n plus 1 over 6 and summation i equal 1 to n of i cubed is n squared times n plus 1 squared over 4 now there are other summations you can get the summation i equal 1 to n of i to the 4 the 5th etc um those we usually don't have as many problems that involve them.
02:49
So they're not ones that are actually stated out, but you could always google search them and see what those are if you need them in a particular setting.
02:59
So let's come back up to the problem we are asked.
03:02
Now we always look to see if there is an exponent other than one outside the parentheses of the multi -term expression.
03:10
If so, you would need to multiply that out before continuing on.
03:15
But here it's just the xonferencees of the multi -term expression.
03:17
Exponent of one.
03:18
So we just have our summation of our two different terms.
03:22
So we'll write this as the summation i equal 1 to 25 of i cubed minus the summation i equal 1 to 25 of 2 i.
03:36
So we applied that top rule.
03:40
Next up in the second term in the second summation, we have two times i, a number times my expression in eyes and with that i can factor that common factor out just of the summation that's involved in so i have the summation i equal 1 to 25 of i cubed minus two times the summation i equal 1 to 25 of i now i have the summation i equal 1 to 25 of i cubed i've got its way to get the total faster by the rule that's the very bottom one we wrote out, and then minus two times the summation i equal 1 to 25 of i, that is the third one from the bottom.
04:32
Now here notice, there's no summation on the right hand side because this takes us to the total we would get if we were to expand out the summation.
04:41
And the n that's involved in the formula is the tapout number from your summation.
04:46
And we checked, and we are starting at 1 and going to that top out, so these rules apply.
04:52
Directly so for the very first summation it's got the i cubed so it's going to follow this bottom rule 25 is my topout number so it's 25 squared times 25 plus 1 squared over 4 then minus two times for i it's n times n plus 1 over 2 my n is 25 so 25 times 25 plus 1 plus 1 over two...