00:01
Okay, so we are going to be using the identity where we have i plus 1 to the 5th minus i to the 5th is equal to 10 i is equal to 5i to the 4th plus 10 i cube plus 10 i squared plus 5i plus 1.
00:18
So we start off using our identity and we then sum both sides as we do every see here.
00:24
And then we can reform the sum on the left hand side and then simplify the right hand side using linearity.
00:31
So we have, well, the left -hand side here, we get a 2 to the 5th, minus 1 to the 5th, plus 3 to the 5th minus 2 to the 5th, right, all the way up to m plus 1 to the 5th minus n to the 5th is equal to, well, the sum of 5i to the 4th plus the sum of 10 i cubed and so on.
00:50
We then can bring the coefficients out in front of the sum.
00:53
So what we end up with is a n plus 1 to the 5th, minus 1.
01:01
To the 5th is equal to 5 times the sum where we have i going from 1 to n of i to the 4th and then plus 10 times the sum where we have i going from 1 to n of i squared and then plus 5 times the sum where we have i going from 1 to n of i going from 1 to n of just 1.
01:34
Okay, so now we can reorder these terms to isolate our i to the fourth term and then make use of the formulas that we know, right, for the sum of i equals 1, i equals 1 to end of i.
01:46
That's just n times m plus 1 over 2.
01:48
And the sum of squares, that's k equals or i equals 1 to n of i squared.
01:53
That's n times m plus 1 times m plus 2 over 6...