Question

If the coefficient of (2r + 4)th term and (r – 2)th term in the expansion of are equal, then the value of r is given by

Answer 1

Using binomial expansion, it would be a right pain to write it all out, faffing about with the powers of x and what not, so it's been relatively nice and only asked about the coefficients (the numbers multiplied by the x terms).  

So really, all you need is to copy out the row from Pascal's Triangle that starts with 1, 18, 153, etc.  

From that, you can see that you've got 19 numbers that would be used in your sequence (counting the 1's at the beginning and end).  

Looking at the r equations, you could, but you don't have to, form some inequalities that limit r and give you an idea of some values to start with.  

2r + 4 < 19, because that's the maximum number of values you've got to work with.  

Therefore, r < 7.5  

r - 2 > 0, because you don't want to end up going in to negative values or the 0th term (I don't even know if that's even possible, just, no, I can't even think about it without getting a headache)  

Which gives you: r > 2 (which is useful-ish, because without it, I would have started with r = 1)  

Using trial and error, with any valid r values you get something along the lines of:  

When r = 3:  

(2r+4)th term = 10th term = 48610 (using pascal's)  

(r-2)th = 1st = 1  

So, obviously, not that one. (You can do r = 4 and 5 as well, but I'll just skip straight to it)  

When r = 6:  

(2r + 4)th term = 16th term = 816  

(r - 2)th term = 4th term = 816  

So, r =6.  

If it's not really that clear feel free to comment/message me so I can clear it up a bit more for you