Prove by mathematical induction that, 2^2+5^2+8^2+...to n terms=n/2(6n^2+3n-1) Share with your friends Share 0 Varun Rawat answered this Let Pn be the statement given byPn : 22 + 52 + 82 + .........+3n-12 = n26n2 + 3n - 1Now, P1 : 3×1-12 = 4 and 12612+3×1-1 = 126+3-1 = 82 = 4So, P1 is true.Let Pm be true. Then,Pm : 22 + 52 + 82 + .........+3m-12 = m26m2 + 3m - 1 ....1We shall now prove that Pm+1 is true. For this, we will show that : 22 + 52 + 82 + .........+3m-12 + 3m+22 = m+126m+12 + 3m+1 - 1Now, 22 + 52 + 82 + .........+3m-12 + 3m+22 =m26m2 + 3m - 1 + 3m+22 using 1=m26m2 + 3m - 1 + 9m2 + 4 + 12m=6m3 + 3m2 - m + 18m2 + 8 + 24m2=6m3 + 21m2 + 23m + 82=6m3+15m2+6m2+8m+15m+82=6m3+6m2+15m2+15m+8m+82=6m2m+1+15mm+1 + 8m+12=m+16m2 + 15m + 82=m+16m2 + 12m + 6 + 3m + 22=m+16m2+2m+1 + 3m + 3 - 12=m+16m+12 + 3m+1 - 12So, Pm+1 is true.Now, Pm is true ⇒Pm+1 is true.Hence, by PMI, the given result is true ∀ n∈N. 1 View Full Answer