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Vanyuwa [196]
3 years ago
10

Use mathematical induction to show that 4^n ≡ 3n+1 (mod 9) for all n equal to or greater than 0

Mathematics
1 answer:
cestrela7 [59]3 years ago
3 0
When n=0, you have

4^0=1\equiv3(0)+1=1\mod9

Now assume this is true for n=k, i.e.

4^k\equiv3k+1\mod9

and under this hypothesis show that it's also true for n=k+1. You have

4^k\equiv3k+1\mod9
4\equiv4\mod9
\implies 4\times4^k\equiv4(3k+1)\mod9
\implies 4^{k+1}\equiv12k+4\mod9

In other words, there exists M such that

4^{k+1}=9M+12k+4

Rewriting, you have

4^{k+1}=9M+9k+3k+4
4^{k+1}=9(M+k)+3k+3+1
4^{k+1}=9(M+k)+3(k+1)+1

and this is equivalent to 3(k+1)+1 modulo 9, as desired.

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Monthly sales of an SUV model are expected to increase at the rate of S′1t2 = -24t 1>3 SUVs per month, where t is time in mon
uysha [10]

The company will continue to manufacture this model for 19 months.

The question is ill-formatted. The understandable format is given below.

An SUV model's monthly sales are anticipated to rise at a rate of

S'(t)=-24^{1/3} SUVs each month, where "t" denotes the number of months and "S(t)" denotes the monthly sales of SUVs. When monthly sales of 300 SUVs are reached, the business intends to discontinue producing this model.

Find S if monthly sales of SUVs are 1,200 at time t=0 (t). How long will the business keep making this model?

Given S'(t)=-24^{1/3} ... ... (1)

Condition (1) at t=0; s(t) =1200

We find S(t)=300 then t=?

a) We find S(t)

from S'(t)=-24^{1/3}

\implies \frac{dS(t)}{dt}=-24t^{1/3} ~~[\because X'(t)=\frac{dX}{dt} \\\implies dS(t) = -24t^{1/3}dt\\

On Integrating both sides

S(t)=-18t^{4/3}+c ~...~...~(2)

Now, at t=0 then S(t)=1200

So, from (2)

1200=0+c

⇒c=1200

\thereforefrom eq (2)

S(t)=-18t^{4/3}+1200 ~...~...~(3)

b) Considering that the firm intends to cease production of this model once monthly sales exceed 300 SUVs.

So, take S(t)=300

from eq (2) 300=-18t^{4/3}+1200

t^{4/3}=\frac{1200-300}{18}\\=\frac{900}{18}=50\\\implies t=50^{3/4}\\\implies t=18.8030

Hence the company will continue 18.8020 months.

Learn more about integration here-

brainly.com/question/18125359

#SPJ10

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They increased the iPad price by $500.

Subtract the total amount from the iPad price:

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The actual price of 10 iPads, without an increase is $4,500

Now divide that by 10 to get the amount for one iPad:

$4,500 ÷ 10 = $450

One iPad actually costs $450 to make.
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