Question

Using mathematical induction, prove that
12 + 22 +32 + ...... +n?
n(n+1)(2n+1) vneN
6​

Answer 1

Step-by-step explanation:

The function of your subconscious mind is to store and retrieve data. Its job is to ensure that you respond exactly the way you are programmed. Your subconscious mind makes everything you say and do fit a pattern consistent with your self-concept. This is your “Master Program.”Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty. 

Math

Answer 2

Answer:

Step-by-step explanation:

To prove :

4 + 8 + 12 + ...+ 4n = 2n(n + 1)

proof :

Let the Given statement be p(n)

p(n): 4 + 8 + 12 + ...+4n = 2n(n + 1)

For n = 1

P(1) : 4 = 2 × 1(1 + 1) = 2 × 2 = 4

So,

p(1) is true when n = 1 .

Now, Let us assume that p(n) is true for some positive intiger k

p(k): 4 + 8 + 12 +...+ 4k = 2k(k + 1) ...(1)

Now , we need to prove that p(k + 1) is also true.

p(k + 1): 4 + 8 + 12 + .... + 4k + 4(k +1) = 2(k + 1)(k + 2)

LHS :

= 4 + 8 + 12 + ...+ 4k + 4(k + 1)

= 2k(k + 1) + 4(k + 1)

= 2k² + 2k + 4k + 4

= 2k² + 6k + 4

RHS :

= 2(k + 1)(k + 2)

= 2[k(k + 2) + 1(k + 2)]

= 2[k² + 2k + k + 2]

= 2(k² + 3k + 2)

= 2k² + 6k + 4

So,

p(k + 1) is true when p(k) is true.

By principal of mathematical induction, statement p(n) is true for all positive intigers