Evaluate -10+ -8 x -1

Prove the following by induction. In each case, n is apositive integer.<br> 2^n ≤ 2^n+1 - 2^n-1 -1.

frutty [35]

<h2>Answer with explanation:</h2>

We are asked to prove by the method of mathematical induction that:

$2^n\leq 2^{n+1}-2^{n-1}-1$

where n is a positive integer.

Let us take n=1

then we have:

$2^1\leq 2^{1+1}-2^{1-1}-1\\\\i.e.\\\\2\leq 2^2-2^{0}-1\\\\i.e.\\2\leq 4-1-1\\\\i.e.\\\\2\leq 4-2\\\\i.e.\\\\2\leq 2$

Hence, the result is true for n=1.

Let us assume that the result is true for n=k

i.e.

$2^k\leq 2^{k+1}-2^{k-1}-1$

Now, we have to prove the result for n=k+1

i.e.

<u>To prove:</u> $2^{k+1}\leq 2^{(k+1)+1}-2^{(k+1)-1}-1$

Let us take n=k+1

Hence, we have:

$2^{k+1}=2^k\cdot 2\\\\i.e.\\\\2^{k+1}\leq 2\cdot (2^{k+1}-2^{k-1}-1)$

( Since, the result was true for n=k )

Hence, we have:

$2^{k+1}\leq 2^{k+1}\cdot 2-2^{k-1}\cdot 2-2\cdot 1\\\\i.e.\\\\2^{k+1}\leq 2^{(k+1)+1}-2^{k-1+1}-2\\\\i.e.\\\\2^{k+1}\leq 2^{(k+1)+1}-2^{(k+1)-1}-2$

Also, we know that:

$-2$

(

Since, for n=k+1 being a positive integer we have:

$2^{(k+1)+1}-2^{(k+1)-1}>0$ )

Hence, we have finally,

$2^{k+1}\leq 2^{(k+1)+1}-2^{(k+1)-1}-1$

Hence, the result holds true for n=k+1

Hence, we may infer that the result is true for all n belonging to positive integer.

i.e.

$2^n\leq 2^{n+1}-2^{n-1}-1$ where n is a positive integer.

6 0

3 years ago

9x -1 = 80<br> A -10<br> B 9<br> C 10<br> D -9

Fittoniya [83]

Answer:

The answer is B!

Step-by-step explanation:

Add 1 to both sides, then divide 80 by 9!

5 0

2 years ago

Solve for pizza in the pie chart!

solniwko [45]

Answer:

24

Step-by-step explanation:

30% of the pie chart is pizza. 30% of 80 = 0.3 * 80 = 24

6 0

2 years ago

Read 2 more answers

What does P(B/A) = 0.50 mean in terms of the Venn diagram?

Serggg [28]

Answer:

of the 4 people that have a sister, 2 of them have a brother

Step-by-step explanation:

probability of B given A

given that the person has a sister, what is the probability they have a brother

P (B/A) = 1/2

with this venn diagram

of the 4 people that have a sister, 2 of them have a brother

3 0

3 years ago

Un empresario contrata a 16 obreros para realizar un trabajo y a cada uno le paga 385 €. Si el empresario quiere ganar 892 €. ¿C

OverLord2011 [107]

Answer:

7052€

Step-by-step explanation:

Si el empresario quiere gana 892€, primero tiene que encontrar el valor total a pagar a los 16 obreros y sumarle a ese valor la ganancia que quiere obtener:

Valor total a pagar a los obreros: 385€*16= 6160€

Valor a cobrar por el trabajo: 6160+892= 7052€

De acuerdo con esto, si el empresario quiere ganar 892 €, debe cobrar 7052€ por el trabajo.

5 0

3 years ago