did anybody know the answer of these questions? urgent!!
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Answer: Mark me as a BRILLIANT ,thankyou.
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
mark me as a BRILLIANT
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no bro I'm from Tamil Nadu u girl or boy
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