Question

prove by using the principle of mathematical induction 1.2+3.4+5.6+.........(2n-1)(2n)=n(n+1)(4n+1)/3
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Answer 1

Step-by-step explanation:

We shall prove the result by principle of mathematical induction.

checking for n=1,

LHS:1.2=2

RHS:

3

1

×1×2×3=2.

Hence true for n=1

Let us assume the result is true for n=k ie.,

1.2+2.3+.....k(k+1)=

3

1

×k×(k+1)×(k+2)

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

that is, to prove 1.2+2.3.....+k(k+1)+(k+1)(k+2)=

3

1

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

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

=

3

1

×k×(k+1)×(k+2)+(k+1)(k+2)

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

3

1

(k+1)]

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

3

1

=RHS.

Hence the result holds for n=k+1.

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