B

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.

_______________________________

Please scroll left - right to view the full solution.

8 0

2 years ago

I’ve looked at this for a while, can someone clear up the answers for me? I can’t figure it out.

navik [9.2K]

What answers? I don't see any

4 0

3 years ago

The Paulsens ordered $75.41 worth of Chinese food. They paid 11.5% sales tax and left a 15% tip on $75.41. What was the total co

mafiozo [28]

75.41*11.5= 8.67
75.41+8.67=84.08
84.08*15%=12.61
84.08+12.61= 96.69
The total is $96.69

8 0

3 years ago

What are the x intercepts of the graph of the function below y=x^2-2x-15

lianna [129]

Answer:

(5, 0) and (-3, 0)

Step-by-step explanation:

To find the x-intercepts, let's set y = 0.

0 = x² - 2x - 15

0 = (x - 5)(x + 3) -- To factor x² - 2x - 15, we need to find 2 numbers with a sum of -2 and product of -15; these numbers are -5 and 3.

x - 5 = 0 or x + 3 = 0 -- Use ZPP (Zero Product Property)

x = 5, x = -3

3 0

4 years ago

Read 2 more answers

Write the equation of each line in slope intercept form (If possible please show work)

Igoryamba

Answer:

y= -2/3 x - 9

Step-by-step explanation:

The slope intercept form is

y = mx+b where m is the slope and b is the y intercept

y = -2/3 x+b

We have a point to substitute into the equation

-5 = -2/3(-6) +b

-5 = 4 +b

Subtract 4 from each side

-5-4 = 4-4+b

-9 = b

y= -2/3 x - 9

8 0

3 years ago