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Sever21 [200]
2 years ago
14

Use the binomial theorem to show that 7^n + 2 is divisible by 3, where n is a positive integer.

Mathematics
1 answer:
masha68 [24]2 years ago
7 0

By the binomial theorem,

7^n=(6+1)^n=6^n+n6^{n-1}+\dbinom n26^{n-2}+\cdots+\dbinom n{n-2}6^2+n6^1+1

Each term containing a factor 6 is divisible by 3, since 3 divides 6. The remaining 1 and 2 sum to 3, which is also divisible by 3.

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f the price charged for a candy bar is​ p(x) cents, where p (x )equals 162 minus StartFraction x Over 10 EndFraction ​, then x t
andreev551 [17]

Answer:

a. 1620-x^2

b. x=810

c. Maximum value revenue=$656,100

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(a) Total revenue from sale of x thousand candy bars

P(x)=162 - x/10

Price of a candy bar=p(x)/100 in dollars

1000 candy bars will be sold for

=1000×p(x)/100

=10*p(x)

x thousand candy bars will be

Revenue=price × quantity

=10p(x)*x

=10(162-x/10) * x

=10( 1620-x/10) * x

=1620-x * x

=1620x-x^2

R(x)=1620x-x^2

(b) Value of x that leads to maximum revenue

R(x)=1620x-x^2

R'(x)=1620-2x

If R'(x)=0

Then,

1620-2x=0

1620=2x

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810=x

x=810

(C) find the maximum revenue

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R(810)=1620x-x^2

=1620(810)-810^2

=1,312,200-656,100

=$656,100

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3 years ago
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