Prove by using PMI: 1+5+9+ +(4n-3) = n(2n-1) for all natural numbers n - Maths - Principle of Mathematical Induction

Question

Prove by using PMI:
1+5+9+ +(4n-3) = n(2n-1) for all natural numbers n

Neha Sethi · Accepted Answer

Dear student

Let Pn be the statement given by Pn:1+5+9+
+4n-3=n2n-1Step I: We have,P1:1=121-1as 1=1So, P1 is trueStep II: Let Pm be true
Then,1+5+9+ +4m-3=m2m-1  
iWe wish to show that Pm+1 is true
 For this we have to show that1+5+9+
+4m-3+4m+1-3=m+12m+1-1Now consider, 1+5+9+
+4m-3+4m+1-3m2m-1+4m+1-3     using i2m2-m+4m+12m2+3m+12m2+2m+m+12mm+1+1m+12m+1m+1=m+12m+1-1So, Pm+1 is trueThus, Pm is true⇒Pm+1 is trueHence by PMI the given result is true for all n∈N
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Prove by using PMI: 1+5+9+..+(4n-3) = n(2n-1) for all natural numbers n

Prove by using PMI:
1+5+9+..+(4n-3) = n(2n-1) for all natural numbers n