Does anyone have a good statement for the end of induction that is applicable to all types of proof? Thanks
If [the summation formula is true/ f(n) is divisible by X/ the general statement is true/ the matrix equation is true (select the appropriate option)] when n=k, then it has been shown to be true for n=k+1. As it is also true for n=1, it must be true for all real, positive integers
Does anyone have a good statement for the end of induction that is applicable to all types of proof? Thanks
"We have shown our statement being true for n=k implies that our statement is true for n=k+1, since we've shown n=1 is true then our statement is true for all natural n."
Which is basically the same thing but in the other direction. I find it much easier to just draw it and find the angle that it makes with the x-axis and then find the total rotation.
1.) I can't quite get this one 2.) sqrt(3-6i) 3.) 4n-4n^2 4.) not a clue, I've never seen anything like this before, some help would be appreciated. 5.)x=1,x=(3-i),x=(3+i),x=(-1-2i),x=(-1+2i) [is it realistic we will get a quintic in the exam?]
need a break after solving that quintic, let me know how I've done if you have some time
Good effort
1). Solution to this one is at the bottom of this page on my site. 2). Not quite. The square root doesn't quite let you add the respective components like that. Try doing each one by one by letting (a+bi)2=2−7i for the first one, then do the second one and add together the results. 3). Not quite! Simplify the summand to get 8−8r, then use the standard rules of summation to get the answer. 4). No problem! Don't worry too much about this one. This problem is sparsely written and beyond the scope of regular FP1. z is any complex number, and the locus is any z that satisfies that equation. 5). Yep, good. No you're probably not going to get a quintic, but it's great practice for your algebra skills. 6). Simple matrix algebra here. a). tests matrix multiplication, then for b) right-multiply both sides by A−1 and see what you get. For c), invert the matrix and verify your result from b). 7). Draw this one on a graph - apply the transformation to the identity matrix (i.e. the points (1,0) and (0,1)) on a 2D graph to get a hint.
Edexcel-specific questions:
8). Hint: Look back on the definition of linear interpolation, especially a graphical one. Check out the formula for approximating α and try rearranging it. PM for the detailed answer to this one. 9). a). Write out the Newton-Raphson formula for this curve and it'll become obvious why 0 is not a good starting point. b). Hint: Look closely at f′(x). What happens to it as x increases? 10). a). Standard FP1 coordinate systems question. Sub the given formula for the straight line y into the curve C's equation and solve to find the points. b). Again, standard question. Convert into Cartesian form (standard rules) and differentiate w.r.t. to x. Then try subbing in the initial parametrised formula for x. 11). a). Simple bookwork. b). Sub the provided straight line equation into the equation of the xy=1 curve to get x and then use the initial formula to find the respective ys for those points. c). Hint: Draw some graphs of the general straight line equation y=kx for a few values of k on the same diagram you did for a). The set of pairs of intersection points a and b for each k were the points you gave for b).