Find an equation that satisfies the given conditions through (1, -5); parallel to the line x+2y=6

Use induction to show that 12 + 22 + 32 + ... + n2 = n(n+1)(2n+1)/6, for all n > 1.

dlinn [17]

Answer with Step-by-step explanation:

Let $P(n)=1^2+2^2+3^2+.....+n^2=\frac{n(n+1)(2n+1)}{6}$

Substitute n=2

Then $P(2)=1+2^2=5$

$P(2)=\frac{2(2+1)(4+1)}{6}=5$

Hence, P(n) is true for n=2

Suppose that P(n) is true for n=k >1

$P(k)=1^2+2^2+3^2+...+k^2=\frac{k(k+1)(2k+1)}{6}$

Now, we shall prove that p(n) is true for n=k+1

$P(k+1)=1^2+2^2+3^2+...+k^2+(k+1)^2=\frac{(k+1)(k+2)(2k+3)}{6}$

LHS

$P(k+1)=1^2+2^2+3^2+.....+k^2+(k+1)^2$

Substitute the value of P(k)

$P(k+)=\frac{k(k+1)(2k+1)}{6}+(k+1)^2$

$P(k+1)=(k+1)(\frac{k(2k+1}{6})+k+1)$

$P(k+1)=(k+1)(\frac{2k^2+k+6k+6}{6})$

$P(k+1)=(k+1)(\frac{2k^2+7k+6}{6})$

$P(k+1)=(k+1)(\frac{2k^2+4k+3k+6}{6})$

$P(k+1)=\frac{(k+1)(k+2)(2k+3)}{6}$

LHS=RHS

Hence, P(n) is true for all n >1.

Hence, proved

4 0

3 years ago

Find the slope will give brainliest.

crimeas [40]

Answer:

The slope is 0.

Step-by-step explanation:

$m=\frac{rise}{run}=\frac{2-2}{2-7}=\frac{0}{-5}=0$

Hope this helps.

8 0

3 years ago

Which of the following is a linear function? (89 POINTS)HURRY PLSSSSS

Dennis_Churaev [7]

<h3>Solution:</h3>

A linear function is a straight line formed on a coordinate plane.

To classify an equation as a linear function:

It must not have any exponent in the equation.
If there is a fraction, the denominator of that fraction must not multiply by x.

If any of these two occurs in an equation, the line can form a curve in a line, which is a non-linear function. A non-linear function is a curved line formed on a coordinate plane.

Option A - y = x³ + 1:

This equation has an exponent in the variable x. Hence, this is a non-linear function.

Option B - 5x/6 = y - 4:

This equation has no exponents in the equation. Hence, this is a linear function and also our answer.

Option C - y = 8x²

This equation has an exponent in the term 8x². Hence, this is a non-linear function.

Option D - y = 1/5x

The denominator of the fraction on the RHS side is multiplied by x. Hence, this is not a linear function.

Hence, Option B is correct.

Learn more about linear functions: brainly.com/question/15253497

8 0

2 years ago

Find the measure of each angle listed below.

gogolik [260]

Answer:

<P 40, <Q 25, <PRQ 115, <PRS 65

Step-by-step explanation:

exteririor angle = sum of remote angles

(3x + 4) + (2x + 1) = (6x - 7)

3x + 2x + 4 + 1 = 6x - 7

5x + 5 = 6x - 7

+7 +7

5x + 12 = 6x

-5x -5x

12 = x

x = 12

find angle mesures:

3(12) + 4

36 + 4

40

2(12) + 1

24 + 1

25

6(12) - 7

72 - 7

65

forms linear pair

65 + m<PRQ = 180

-65 -65

m<PRQ = 115

3 0

3 years ago

Choose all below that correctly show addition with models.

stiks02 [169]

The first second and third

6 0

3 years ago

Read 2 more answers