If x is odd, then 2^x + 3^x is also odd

  • Thread starter Thread starter epkid08
  • Start date Start date
epkid08
Messages
264
Reaction score
1

Homework Statement



Prove: If x is a positive odd integer, then 2^x + 3^x is also a positive odd integer.


The Attempt at a Solution



Assume x is odd; there exists an integer k such that x = 2k + 1. Then
2^x + 3^x = 2^{2k+1} + 3^{2k+1}

I don't know how to go about simplifying it from here.
 
Physics news on Phys.org


Is 2^x even or odd? What about 3^x? I'm a bit confused why they are specifying x odd. Isn't it true even if x is even?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
View full post »

Similar threads

Odd extension: Show that if [K(a) : K] is odd, then K(a) = K(a^2)
Replies
7
Views
1K
How should I show the following by using the signum function?
Replies
14
Views
2K
Show that k is an odd integer, except when k=2
Replies
9
Views
2K
Prove algebraically that ##n^3+3n-1## is odd
Replies
16
Views
2K
Prove that ##\lim\limits_{x \to \infty} f(x) = 0##
Replies
3
Views
585
Proving that a square of an odd integer is also odd
Replies
3
Views
2K
Cryptography problem involving prime numbers
Replies
5
Views
1K
Separable first order ODE involving tangent
Replies
2
Views
2K
Prove that ## 1^{p}+2^{p}+3^{p}+\dotsb +(p-1)^{p}\equiv 0\pmod {p} ##.
Replies
6
Views
3K
Examples of Partitions: How to Divide Nonzero Integers into Infinite Sets?
Replies
5
Views
2K

Hot Threads

Recent Insights

Back
Top