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horrorfan [7]
3 years ago
13

Linear, exponential or neither f(1) = 13, f(n)=6f(n-1)

Mathematics
1 answer:
NikAS [45]3 years ago
8 0
↓The correct answer is here↓

brainly.com/question/7666093
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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
How do you get the perimeter of a triangle, all side are 28 cm?
Alisiya [41]

Perimeter of the triangle, a+b+c

In this case, a=b=c=28

So, perimeter would be 28*3 = 84 cm

7 0
3 years ago
Mrs.Ramirez bought 2 qt of orange juice. If the juice is served in 6-oz glasses,how many glasses can be completely filled?
alekssr [168]
Hey there!

<span>There is 32 oz per qt. 2qts=64oz. Divide 64 by 6. </span><span> 6 goes evenly into 64 10 times with a remainder. So the answer is 10 glasses. 
</span>
Hope this helps!
4 0
3 years ago
−20=−4x−6x Find the value of x
xeze [42]

Answer:

\huge\boxed{x=2}

Step-by-step explanation:

-20=-4x-6x\qquad|\text{combine like terms}\\\\-20=-10x\qquad|\text{divide both sides by (-10)}\\\\\dfrac{-20}{-10}=\dfrac{-10x}{-10}\\\\2=x\to x=2

4 0
3 years ago
If there are 400 cubic centimeters of a chemical in 1 liter of solution, how many cubic centimeters of water must be added to di
mars1129 [50]
1l=1dm^3\\1dm=10cm\to1dm^3=(10cm)^3=1000cm^3\\\\1l=1000cm^3\\-------------\\  \begin{array}{ccc}400cm^3&-&25\%\\1000cm^3+x&-&100\%\end{array}\\cross\ multiply\\25(1000+x)=400\times100\\25000+25x=40000\ \ \ \ \ |subtract\ 25000\ from\ both\ sides\\25x=15000\ \ \ \ |divide\ both\ sides\ by\ 25\\\boxed{x=600\ (cm^3)}\\\\Answer:Must\ be\ added\ 600cm^3\ of\ water.
5 0
3 years ago
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