00:02
Okay, we've got this arithmetic progression.
00:07
We really only need the first three terms.
00:10
They said the first term is a natural number, so it's called n.
00:14
And then the common difference is a whole number.
00:18
So the next term would be n plus a.
00:21
If we let a be our common difference, then be n plus 2a.
00:26
Then they say if we subtract 2 from the first term and add, 10 to the last term, then we'll have a geometric progression.
00:43
And we want to know which of their four choices could the common difference a equal.
00:52
So one thing i know is that if we square that middle term, it's going to be the same as multiplying that first term and that last term, because it's the geometric mean of too.
01:19
So maybe if we look at that product, maybe we'll get something meaningful.
01:26
So this would be n squared plus 2a n plus a plus a squared over here.
01:39
And then we get n squared plus 2a n plus 10n minus 2n minus 4a minus 20.
01:56
Then there's some things we can cancel because they're shared on both sides, like the n squared and the 2a .n.
02:05
So now we've got a squared is what we combine these two to get 8n minus 4a minus 20.
02:20
Okay, so then i notice that i can factor out a 4.
02:29
So i'll get 2n minus a minus.
02:36
So whatever my common difference a is, i know that when i square it, i will get a multiple of 4.
02:47
So let's take a look at their answer choices here.
02:51
So let's say, well, what if a is 4? so if a is 4, then 1 1 squared equals 4 times 2n minus 4 minus 5.
03:07
So 16 would equal 4 times 2n minus 9.
03:28
Okay, so notice that this would have to be an odd number.
03:34
So when i divide 16 by 4, i get 4, which is not an odd number.
03:42
So a cannot be 4.
03:44
So let's check some of the other options.
03:47
A is 8.
03:53
Well, let's see.
03:53
We'd get 8 squared equals 4 times 2n minus 8 minus 5.
04:02
So 64 is 4.
04:06
Let's see, 8 plus 5 is 13.
04:13
So again, this is an odd number...