Using mathematical induction prove that 2+7+12+....+(5n-3)=n (5n-1)/2.

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Please find below the solution to the asked query:

Let Pn :2+7+12....+5n-3=n5n-12For n=1L.H.S = 5×1-3 = 5 - 3 = 2R.H.S= 1.5.1-12=42=2 L.H.S=R.H.SSo, P1 is true.Let Pk is true.Pk : 2+7+12....+5k-3=k5k-12Now, 2+7+12....+5k-3+5k+1-3 =k5k-12 + 5k+2 = k5k-1 + 10k+42 =5k2-k+10k+42=5k2+9k+42=5k2+5k+4k+42=5kk+1+4k+12=k+15k+1-12So, Pk+1 is true whenever Pk is true.Hence, P1 and Pk+1 is true when Pk is true.Hence by the PMI, Pn is true for all n  N.

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