prove 3x+6x+9x+ +to n terms= 3/2n(n+1)x by PRINCIPLE OF MATHEMATICAL INDUCTION

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We havePn:3x+6x+9x+....+3nx=32nn+1x nth term will nth multiple of 3Step 1: Base CaseFor n=1, we have3x=321+1x3x=3xHence Pn is true for n=1Step 2: Inductive hypothesisLet Pn be true for n=k i.e.3x+6x+9x+....+3kx=32kk+1x ;equationiStep 3: Inductive CaseConsider3x+6x+9x+....+3kx+3k+1x=32kk+1x+3k+1x Using i=3k+1x.k2+1=3k+1x.k+22=32k+1k+1+1xHence Pn is true for n=k+1.Since Pn is true for n=1 and when Pk is true then Pk+1 is also true, henceby principle of mathematical induction, Pn is true for all nN

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