IfS1,S2andS3 be the sum of n, 2n, 3n terms respectively of an AP prove

1.	IfS1,S2andS3 be the sum of n, 2n, 3n terms respectively of an AP prove that 3S1-3S2+S3=0
Answer» {tex}{S_1} = \\frac{n}{2}\\left[ {2a + (n - 1)d} \\right]{/tex}{tex}{S_2} = \\frac{{2n}}{2}\\left[ {2a + (2n - 1)d} \\right]{/tex}{tex}{S_3} = \\frac{{3n}}{2}\\left[ {2a + (3n - 1)d} \\right]{/tex}R.H.S = 3(S2 - S1){tex} = 3\\left[ {\\frac{{2n}}{2}(2a + (2n - 1)d - \\frac{n}{2}(2a + (n - 1)d)} \\right]{/tex}{tex} = 3\\left[ {\\frac{n}{2}\\left[ {4a + 4nd - 2d - 2a - nd + d} \\right]} \\right]{/tex}{tex} = 3\\left[ {\\frac{n}{2}(2a + 3nd - d)} \\right]{/tex}{tex} = \\frac{{3n}}{2}\\left[ {2a + (3n - 1)d} \\right] = {S_3}{/tex}

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