Mathematics, 06.09.2019 17:20 emmie50

Prove that 9 divides 4" + 6n – 1 for n > 1 by mathematical induction.

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Answer from: wolfiewolffromsketch

This is a classic problem I think that you mean $4^{n}+6n-1$

This problem clearly states to use proof by induction, follow the complete answer below.

Step-by-step explanation:

To prove something by induction we have to make a proposition first, in this case:

$P(n): 4^{n}+6n-1$ is divisible by 9

Where $n \in \mathbb{N}$ is the order of the proposition.

First we have to prove the first few orders, let's just check P(1):

$4^1+6\times1-1=9$ which is clearly divisible by 9

Now let's assume that P(n) is true that means, our original proposition is true. Now let's try to find out whether P(n+1) is true:

$4^{n+1}+6(n+1)-1=4\cdot 4^n+6n+5=4\cdot n+\underbrace{4\cdot 6n-18n}_{6n}-\underbrace{4\cdot 1+9}_{5}=4(4^n+6n-1)-18n+9=4(4^n+6n-1)-9(2n-1)$

The 4(4^n+6n-1) is divisible by 9 because we assumed that " is divisible by 9" was a true proposition. On the other hand 9(2n-1) has a factor of 9 regardless of n, thus is also divisible by 9 that means that P(n+1) is true and it follows that P(n) must also be true, thus proving our initial statement.

Answer from: Quest

Intersecting secant-tangent theorem states that if a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment.so to break it down: 1. pq acts as the tangent and ps as the secant with an intersection at p.sr = 21rp = 3x + 3pq = 4x + 4following to equation: tangent squared = the two segments multiplied by each otherso (4x + 4) ^ 2 = (3x + 3) • (21) simplifies to 16x^2 - 31x - 47 = 0 or (16x - 47)(x + 1) x = -1 or x = 47/162. pq acts as the tangent and ps as the secant with an intersection at p (again! )sr = 16rp = xpq = 15(15) ^ 2 = 16 • x225 = 16xx = 225/163. the interesting chords are proportional to each other so a ratio is possible to set up: ap dp 10 3 + x = = —— = pc pb 8 xcross multiple to get 10x = (3 + x)(8)x = 12

Answer from: Quest

the answer is 13

step-by-step explanation:

5*5=25

12*12=144

25+144=169

169^2=13

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Prove that 9 divides 4" + 6n – 1 for n > 1 by mathematical induction....

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