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2 years ago

Thank you for your help.

Mathematics

1 answer:

Nonamiya [84]2 years ago

8 0

Answer:

q = -15

Step-by-step explanation:

5q = -75

Divide both sides by 5 : q = -15

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How can i prove this property to be true for all values of n, using mathematical induction.

chubhunter [2.5K]

Proof -

So, in the first part we'll verify by taking n = 1.

$\implies \: 1 = {1}^{2} = \frac{1(1 + 1)(2 + 1)}{6}$

$\implies{ \frac{1(2)(3)}{6} }$

$\implies{ 1}$

Therefore, it is true for the first part.

In the second part we will assume that,

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

and we will prove that,

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

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

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

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

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

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

${1}^{2} + {2}^{2} + {3}^{2} + ..... + {k}^{2} + (k + 1)^{2} = \frac{(k+1)(2k^2+7k+6) }{6}$

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

Henceforth, by using the principle of mathematical induction 1²+2² +3²+....+n² = n(n+1)(2n+1)/ 6 for all positive integers n.

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8 0

2 years ago

What is the next number in the sequence below?

zysi [14]

Answer is
B. 6
It’s dividing by 2

3 0

3 years ago

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Tell me how to shade the area on the grid that shows 7/8x3/4

viva [34]

If the grid is an 8x8:

To solve this we need to convert 3/4 into a fraction x/8 so that they have the same denominator.

To do this we can set a porportion and find a common denominator. 4 x 2 = 8 so the common denominator would be 8 and the rule would be x 2 so 4 x 2 = 8 and 3 x 2 = 6

So we get 7/8 and 6/8

From top to bottom shade 7 spaces in a row

From left to right shade 6 spaces in a row

Fill in the spaces in between to make a square

I hope this helps!

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3 years ago

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The ages of trees in a forest are normally distributed with a mean of 25 years and a standard deviation of 4. Approximately what

harkovskaia [24]

Answer:

78.88%

Step-by-step explanation:

We have been given that

$\mu=25,\sigma=4,x_1=20,x_2=30$