What is the value of the function y=2x−3 when x=−1 ?

Show that 3 · 4^n + 51 is divisible by 3 and 9 for all positive integers n.

Leni [432]

Answer:

To prove that 3·4ⁿ + 51 is divisible by 3 and 9, we have;

3·4ⁿ is divisible by 3 and 51 is divisible by 3

Where we have;

$S_{(n)}$ = 3·4ⁿ + 51

$S_{(n+1)}$ = 3·4ⁿ⁺¹ + 51

$S_{(n+1)}$ - $S_{(n)}$ = 3·4ⁿ⁺¹ + 51 - (3·4ⁿ + 51) = 3·4ⁿ⁺¹ - 3·4ⁿ

$S_{(n+1)}$ - $S_{(n)}$ = 3( 4ⁿ⁺¹ - 4ⁿ) = 3×4ⁿ×(4 - 1) = 9×4ⁿ

∴ $S_{(n+1)}$ - $S_{(n)}$ is divisible by 9

Given that we have for S₀ = 3×4⁰ + 51 = 63 = 9×7

∴ S₀ is divisible by 9

Since $S_{(n+1)}$ - $S_{(n)}$ is divisible by 9, we have;

$S_{(0+1)}$ - $S_{(0)}$ = $S_{(1)}$ - $S_{(0)}$ is divisible by 9

Therefore $S_{(1)}$ is divisible by 9 and $S_{(n)}$ is divisible by 9 for all positive integers n

Step-by-step explanation:

5 0

3 years ago

Tell whether the statement is true or false. Tan20 degrees greater than cot20 degrees

Oxana [17]

True because tan(20)=2.1 and cot(20)=is o.44

8 0

3 years ago

A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n indepe

Naddik [55]

Answer:

P (X ≤ 4)

Step-by-step explanation:

The binomial probability formula can be used to find the probability of a binomial experiment for a specific number of successes. It does not find the probability for a range of successes, as in this case.

The range "x≤4" means x = 0 or x = 1 or x = 2 or x = 3 or x = 4, so there are five different probability calculations to do.

To to find the total probability, we use the addition rule that states that the probabilities of different events can be added to find the probability for the entire set of events only if the events are Mutually Exclusive. The outcomes of a binomial experiment are mutually exclusive for any value of x between zero and n, as long as n and p don't change, so we're allowed to add the five calculated probabilities together to find the total probability.

The probability that x ≤ 4 can be written as P (X ≤ 4) or as P (X = 0 or X = 1 or X = 2 or X = 3 or X = 4) which means (because of the addition rule) that P(x ≤ 4) = P(x = 0) + P(x = 1) + P (x = 2) + P (x = 3) + P (x = 4)

Therefore, the probability of x<4 successes is P (X ≤ 4)

3 0

3 years ago

Find the hypotenuse of an isosceles right triangle when the legs each measure 3\sqrt{x} 2inches.

Tresset [83]

The hypotenuse of the right triangle measures 3 units.

<h3>How to find the hypotenuse?</h3>

Here we know that both legs measure:

$L =\frac{3}{\sqrt{2} }$