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The idea behind the sum of powers of 2 - Mathematics Stack Exchange I know that the sum of powers of 2 2 is 2n+1 − 1 2 n + 1 1, and I know the mathematical induction proof But does anyone know how 2n+1 − 1 2 n + 1 1 comes up in the first place For example, sum of n numbers is n(n+1) 2 n (n + 1) 2 The idea is that we replicate the set and put it in a rectangle, hence we can do the trick What is the logic behind the sum of powers of 2 2 formula?
Differentiating Definite Integral - Mathematics Stack Exchange You'll need to complete a few actions and gain 15 reputation points before being able to upvote Upvoting indicates when questions and answers are useful What's reputation and how do I get it? Instead, you can save this post to reference later
ABCD is a parallelogram, P is any point on AC. Through P, MN is drawn . . . ABCD A B C D is a parallelogram, P P is any point on AC A C Through P P, MN M N is drawn parallel to BA B A cutting BC B C in M M and AD A D in N N SR S R is drawn parallel to BC B C cutting BA B A in S S and CD C D in R R Show that [ASN] + [AMR] = [ABD] [A S N] + [A M R] = [A B D] (where [ ] [ ] denotes the area of the rectilinear figure)
quadratics - Newtons Identity - Mathematics Stack Exchange While reading about quadratic equations, I came across Newton's Identity formula which said we can express αn +βn α n + β n in simpler forms but not given any explanation They wrote Sn = αn +βn S n = α n + β n and plugged in the quadratic equation f(x) = ax2 + bx + c f (x) = a x 2 + b x + c to write:
group theory - $a^ma^n=a^ {m+n}$ and $ (a^m)^n=a^ {mn}$ proof . . . Here is a proof as I allude to in my comments, although this proof depends on having a more rigorous inductive definition of exponentiation as follows: Base case: a0 = Id a 0 = Id, the identity element of the group Inductive case: assuming an a n is defined for some integer n ≥ 0 n ≥ 0, define an+1 =an ⋅ a a n + 1 = a n ⋅ a Then you can prove each identity by induction on one of the
What is the formula for the difference between CI and SI? If principal, time and rate are given how,do I find the difference between Compound interest and Simple Interest? p=12,000 n=1 and a 1 2 yrs r=10% per year Formulae that I know: CI - SI for 2 y
How to add and subtract values from an average? I know that's an old thread but I had the same problem I want to add a value to an existing average without having to calculate the total sum again to add a value to an exisitng average we only must know for how many values the average was calculated for:
How to calculate reflected light angle? - Mathematics Stack Exchange On a two dimensional plane, line X X is at an angle of x x radians and an incoming light travels at an angle of y y radians How can I calculate the angle of the outgoing light reflected off of the line X X? How can I cover all possible cases? Edit: I was trying to figure out Project Euler problem 144