+0

# Equivalent Expressions

0
320
6
+466

Show that replacing k with k + 1 in 1 - 1/2k gives an expression equivalent to 1 - 1/2k + 1/2k+1

#2
+80956
+15

Show that replacing k with k + 1 in 1 - 1/2^k gives an expression equivalent to 1 - 1/2^k + 1/2^(k+1)

1  -  1/ [2^(k + 1)]  =

1 - 1 / [2^k * 2] =

1 - 1/2^k (1/2)  =

1 + (-1/2)(1/2^k) =                     [notice that  -1/2  =  -1  + 1/2]

1 + [ -1 + 1/2] [ 1/2^k]  =

1 - 1/2^k  + (1/2)(1/2^k)  =

1 - 1/2^k + 1 / [ 2^k *2]  =

1 - 1/2^k +  1/ 2^(k + 1)

CPhill  Feb 4, 2016
Sort:

#1
+91435
+10

Show that replacing k with k + 1 in 1 - 1/2k gives an expression equivalent to 1 - 1/2k + 1/2k+1

Going back to front it is easy enough.

$$1-\frac{1}{2^k}+ \frac{1}{2 ^{k+1} } \\ =1-\left[\frac{1}{2^k}- \frac{1}{2 ^{k+1} } \right]\\ =1-\left[\frac{2}{2^{k+1}}- \frac{1}{2 ^{k+1} } \right]\\ =1-\left[\frac{1}{2^{k+1} } \right]\\ =1-\frac{1}{2^{k+1} } \\$$

Now I want to work out how to do it in the other direction ://

Melody  Feb 4, 2016
#2
+80956
+15

Show that replacing k with k + 1 in 1 - 1/2^k gives an expression equivalent to 1 - 1/2^k + 1/2^(k+1)

1  -  1/ [2^(k + 1)]  =

1 - 1 / [2^k * 2] =

1 - 1/2^k (1/2)  =

1 + (-1/2)(1/2^k) =                     [notice that  -1/2  =  -1  + 1/2]

1 + [ -1 + 1/2] [ 1/2^k]  =

1 - 1/2^k  + (1/2)(1/2^k)  =

1 - 1/2^k + 1 / [ 2^k *2]  =

1 - 1/2^k +  1/ 2^(k + 1)

CPhill  Feb 4, 2016
#3
+91435
+10

Show that replacing k with k + 1 in 1 - 1/2k gives an expression equivalent to 1 - 1/2k + 1/2k+1

We can also do this with partial fractions and then it will go in the correct direction.

$$1-\frac{1}{2^{k+1}}\\ =1+\;\;\frac{-1}{2*2^k}\\ \mbox{There are some integers A and B such that}\\ =1+\;\;\frac{A}{2^k}+\frac{B}{2^{k+1}}\\ =1+\;\;\frac{2A}{2^{k+1}}+\frac{B}{2^{k+1}}\\ =1+\;\;\frac{2A}{2^{k+1}}+\frac{B}{2^{k+1}}\\ =1+\;\;\frac{2A+B}{2^{k+1}}\\ \qquad SO\\ \qquad 2A+B=-1\\ \qquad \mbox{One solution to this is }\;\;A=-1\;\;and\;\;B=+1\\ =1+\;\;\frac{-1}{2^k}+\frac{1}{2^{k+1}}\\ =1-\;\;\frac{1}{2^k}+\frac{1}{2^{k+1}}\\$$

[Any A and B such that    2A+B= -1   will also be true ]

Melody  Feb 4, 2016
#4
+91435
+10

Yours is a really nice alternate solution Chris

Melody  Feb 4, 2016
#5
+8623
+5

The Queen as returned!! * bows  *

Hayley1  Feb 4, 2016
#6
+91435
0

Thanks Hayley,

I still do not expect to be as conscientious as before - I still have many other things cluttering up the "to do" list ://

Melody  Feb 4, 2016

### 24 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details