PLEASE SHOW WORK and ANSWER EVERYTHING ASKED Prove that Here is a general outlin

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PLEASE SHOW WORK and ANSWER EVERYTHING ASKED

Prove that Here is a general outline you can use. Basis step. Prove P(1) Induction step. Write out P(k) by replacing "n" with "k" in the original equation. Now replace "k" with "k+1." This is what we need to show. Using the assumption that P(k) is true, replace "1 +2+... +k" in the P(k+1) statement. If necessary, multiply away any constant denominators in the new P(k+1) formulation. Multiply out the left-hand side and the right-hand side to establish the equality.

Explanation / Answer

P(1)

:

LHS:

1/(1*2)=0.5

RHS:

1*1/(1+1)=1/2=0.5

so it satisfies.

assume P(K) is true.

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

then P(k+1)= (1/1.2)+(1/2.3)+.....+(1/k.(k+1)) +(1/(k+1)*(k+2))

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

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

=(k^2+2*k+1)/(k+1)*(k+2)

=(k+1)^2/((k+1)*(k+2))

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

=RHS

hence proved.

note:

LHS:left hand side

RHS:right hand side

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