Question

did anybody know the answer of these questions? urgent!!​

Answer 1

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prove that by 1.2+2.3+3.4+    +n(n+1) = (n(n+1)(n+2))/3

Step-by-step explanation:

checking for n=1,  

LHS:1.2=2

RHS:  1/3 ×1×2×3=2

Hence true for n=1

Let us assume the result  n=k  

1.2+2.3+   k(k+1)= 1/3

×k×(k+1)×(k+2)  

We shall prove the result to be true for n=k+1.  

to prove 1.2+2.3    +k(k+1)+(k+1)(k+2)= 1/3

(k+1)(k+2)(k+3)

consider LHS:1.2+2.3    +k(k+1)+(k+1)(k+2)

= 1/3

×k×(k+1)×(k+2)+(k+1)(k+2)

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

1/3 (k+1)]

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

​=RHS.

Hence the result holds for n=k+1.  

Hence proof is complete by PMI and therefore the result holds.

consider the statement p(x):n(n+1)(2n+1) is divisible by 6

(i) verify the statement for n=2

  Let P(n):  n(n + 1)(n + 2) is divisible by 6.

P(1):  1(1 + 1)(1 + 2) = 6 which is divisible by 6. Thus P(n) is true for n = 1.

Let P(k)  be true for some natural number k.

P(k):  k(k + 1)(k + 2) is divisible by 6.

Now we prove that P(k + 1) is true whenever P(k) is true.

Now, (k + 1)(k + 2)(k + 3) = k(k + 1)(k + 2) + 3(k + 1)(k + 2)

Since, we have assumed that k(k + 1)(k + 2) is divisible by 6, also (k + 1)(k + 2) is divisible by 6 as either of (k + 1) and ( k + 2) has to be even number.

P(k + 1) is true.

Thus P(k + 1) is true whenever P(k) is true.

i hope it may helpfull to you

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Answer 2

Answer:

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