Question

`1+3+3^(2)+…….+3^(n-1) =((3^(n)-1))/(2)`

Answer 1

Let `P (n) :1+3+3^(2)+…..+3^(n-1)=(3^(n)-1)/(2)`
`" If "n =1 , "the " L.H.S. =1`
`R.H.S. =(3^(1)-1)/(2)=(3-1)/(2)=1`
`:. " "L.H.S. =R.H.S.`
Therefore the statement P (n) is true for n=1
Let P (n) be true for n=K.
`:. P (k) : 1+3+3^(2) +.....+3^(k-1) =(3^(K)-1)/(2)`
`P (k+1) :1+3+3^(2)+......+3^(k)`
`=1+3+3^(2)+......+3^(k)`
`=1+3+3^(2)+.......+3^(k-1)+3^(k)`
`=(3^(k)-1)/(2) +3^(k)`
`=(3^(k)-1+2.3^(k))/(2)=((1+2)3^(k)-1)/(2)`
`=(3.3^(k)-1)/(2)=(3^(k+1)-1)/(2)`
Then the statement P (n) is also true for n=K +1 ,
Hence form the principle of mathematical induction P (n) is true for `n in N`