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Aleksandr [31]
2 years ago
12

Use mathematical induction to prove the statement is true for all positive integers n. 1^2 + 3^2 + 5^2 + ... + (2n-1)^2 = (n(2n-

1)(2n+1))/3)
Mathematics
1 answer:
Charra [1.4K]2 years ago
4 0

Answer:

The statement is true is for any n\in \mathbb{N}.

Step-by-step explanation:

First, we check the identity for n = 1:

(2\cdot 1 - 1)^{2} = \frac{2\cdot (2\cdot 1 - 1)\cdot (2\cdot 1 + 1)}{3}

1 = \frac{1\cdot 1\cdot 3}{3}

1 = 1

The statement is true for n = 1.

Then, we have to check that identity is true for n = k+1, under the assumption that n = k is true:

(1^{2}+2^{2}+3^{2}+...+k^{2}) + [2\cdot (k+1)-1]^{2} = \frac{(k+1)\cdot [2\cdot (k+1)-1]\cdot [2\cdot (k+1)+1]}{3}

\frac{k\cdot (2\cdot k -1)\cdot (2\cdot k +1)}{3} +[2\cdot (k+1)-1]^{2} = \frac{(k+1)\cdot [2\cdot (k+1)-1]\cdot [2\cdot (k+1)+1]}{3}

\frac{k\cdot (2\cdot k -1)\cdot (2\cdot k +1)+3\cdot [2\cdot (k+1)-1]^{2}}{3} = \frac{(k+1)\cdot [2\cdot (k+1)-1]\cdot [2\cdot (k+1)+1]}{3}

k\cdot (2\cdot k -1)\cdot (2\cdot k +1)+3\cdot (2\cdot k +1)^{2} = (k+1)\cdot (2\cdot k +1)\cdot (2\cdot k +3)

(2\cdot k +1)\cdot [k\cdot (2\cdot k -1)+3\cdot (2\cdot k +1)] = (k+1) \cdot (2\cdot k +1)\cdot (2\cdot k +3)

k\cdot (2\cdot k - 1)+3\cdot (2\cdot k +1) = (k + 1)\cdot (2\cdot k +3)

2\cdot k^{2}+5\cdot k +3 = (k+1)\cdot (2\cdot k + 3)

(k+1)\cdot (2\cdot k + 3) = (k+1)\cdot (2\cdot k + 3)

Therefore, the statement is true for any n\in \mathbb{N}.

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marysya [2.9K]
Hello there, and thank you for posting your question here on brainly.

Something I noticed right away is that 4/8 can be simplified.

4/8 ===> 2/4

So now we have 2/4 - 1/4

Just subtract the numerators only.

2/4 - 1/4 = 1/4

1/4 cannot be simplified, so 1/4 is your final answer.

Hope this helped!! ☺♥
5 0
3 years ago
Can someone help me with this answer plsss
solniwko [45]

Answer: 44 ft

Step-by-step explanation:

if you add up the right square it adds up to 16 4+4+4+4 all sides are equal since its a square.

The left rectangle is

8+8=16

6+6=12

16+12=28

Now you add the square on the right to the square on the left which is

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4 0
2 years ago
Which expression results from using the Distributive Property? 2(5 + 3r)
Natasha_Volkova [10]

Answer:

6r + 10

Explanation:

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Artemon [7]

Answer:

The slope is 1.4

Step-by-step explanation:

Have a great day!!!

Kono Dio Da!!!

6 0
3 years ago
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VARVARA [1.3K]

Answer:

C

Step-by-step explanation:

To find (g ○ f)(x), substitute x = f(x) into g(x) , that is

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2 years ago
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