12+14+18+⋯+12n=1−12n.
Let P (n) =12+14+18+⋯+12n=1−12n
For n = 1
P(1)=121=1−121⇒12=frac12
∴ P (1) is true
Let P (n) be true for n = k
∴P(k)=12+14+18+⋯+12k=1−12kFor n = k + 1P(k+1)=12+14+18+⋯+12k+12k+1=1−12k+1=1−]frac12k+12k+1=1−(12k−12k+1)=1−(2−12k+1)=−12k+1
∴ P(k + 1 ) is true
thus P (k) is true ⇒P(k+1) is true
hence by principle fo mathematical induction,
p (n) is true for all nϵN.