Use induction to prove the following formula is true for all integers n where n greaterthanorequalto 1. 1 + 4 + 9 + .. + n^2 = n(n + 1)(2n + 1)/6

Answers

Answer 1
Answer:

Answer with Step-by-step explanation:

Since we have given that

1+4+9+........................+n² = (n(n+1)(2n+1))/(6)

We will show it using induction on n:

Let n = 1

L.H.S. :1 = R.H.S. : (1* 2* 3)/(6)=(6)/(6)=1

So, P(n) is true for n = 1

Now, we suppose that P(n) is true for n = k.

1+4+9+...................+k^2=(k(k+1)(2k+1))/(6)

Now, we will show that P(n) is true for n = k+1.

So, it L.H.S. becomes,

1+4+9+......................+(k+1)^2

and R.H.S. becomes,

((k+1)(k+2)(2k+3))/(6)

Consider, L.H.S.,

1+4+9+..+k^2+(k+1)^2\n\n=(k(k+1)(2k+1))/(6)+(k+1)^2\n\n=k+1[(k(2k+1))/(6)+(k+1)]\n\n=(k+1)[(2k^2+k+6k+6)/(6)]\n\n=(k+1)(2k^2+7k+6)/(6)]\n\n=(k+1)(2k^2+4k+3k+6)/(6)]\n\n=(k+1)[(2k(k+2)+3(k+2))/(6)]\n\n=((k+1)(2k+3)(k+2))/(6)

So, L.H.S. = R.H.S.

Hence, P(n) is true for all integers n.


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This is for my brother’s test What are the measures of L1 and L2? Show your work or explain your answers.

Answers

Answer:

angle 2 is 75°

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Which of the following graphs is a function?

Answers

Answer:

a

Step-by-step explanation:

because of the line test

mimi read 1/4 of a book on wednesday. she read 82 more pages on thursday, and she finished the book on friday by reading the last 98 pages. how many pages were in mimi's book.

Answers

Answer:

240 pages

Step-by-step explanation:

Let x = the TOTAL number of pages in the book. So we'll add up all her reading for the week and it should equal x, the total number of pages.

see image.

4. A sphere with a diameter of 16 mm has the same surface area as the total surface area of a right cylinder with the base diameter equal to the sphere diameter.How high is the cylinder?
O A. 8 mm
OB. 14 mm
O C. 16 mm
O D. 18 mm

Answers

Answer: The answer is A, 8 mm.

Step-by-step explanation:

(c) In how many way we can list the digits{1,1,2,2,3,4,5}so that notwo identical digit are in consecutive positions?

Answers

Answer:

660 ways

Step-by-step explanation:

we have two numbers in consecutive positions in this question

(1,1) and (2,2)

numbers of ways that (1,1) are in consecutive positions = 6!/2! = 360

number of ways that (2,2) are in consecutive positions = 6!/2! = 360

the permutation of (11),(22),3,4,5 = 5!

ps:I countedthepairsasoneeach.

5! = 120

to get total number of permutations

7!/2!2!

= 5040/4

= 1260

the number of ways that 2 identical digits are not consecutively positioned = 1260-360-360+120

= 660 ways

Help math math math

Answers

Answer:

Yes

Step-by-step explanation:

Answer:

The answer is Yes. Hope this helps :)

Step-by-step explanation: