Home
Class 11
MATHS
Using principle of mathematical inductio...

Using principle of mathematical induction, prove that
`1 + 3 + 3^(2) + … 3^(n-1) = (3^(n) - 1)/(2)`

Text Solution

Verified by Experts

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`
Doubtnut Promotions Banner Mobile Dark
|

Topper's Solved these Questions

  • PRINCIPLE OF MATHEMATICAL INDUCTION

    NAGEEN PRAKASHAN ENGLISH|Exercise Exercise 4|37 Videos
  • PERMUTATION AND COMBINATION

    NAGEEN PRAKASHAN ENGLISH|Exercise Miscellaneous Exercise|11 Videos
  • PROBABILITY

    NAGEEN PRAKASHAN ENGLISH|Exercise MISCELLANEOUS EXERCISE|10 Videos

Similar Questions

Explore conceptually related problems

Using the principle of mathematical induction, prove that 1.3 + 2.3^(2) + 3.3^(2) + ... + n.3^(n) = ((2n-1)(3)^(n+1)+3)/(4) for all n in N .

Using the principle of mathematical induction prove that : 1. 3+2. 3^2+3. 3^3++n .3^n=((2n-1)3^(n+1)+3)/4^ for all n in N .

Using the principle of mathematical induction , prove that for n in N , (1)/(n+1) + (1)/(n+2) + (1)/(n+3) + "……." + (1)/(3n+1) gt 1 .

Using the principle of mathematical induction, prove that : 1. 2. 3+2. 3. 4++n(n+1)(n+2)=(n(n+1)(n+2)(n+3))/4^ for all n in N .

By using principle of mathematical induction, prove that 2+4+6+….2n=n(n+1), n in N

By the principle of mathematical induction prove that 3^(2^(n))-1, is divisible by 2^(n+2)

Using the principle of mathematical induction prove that 41^n-14^n is a multiple of 27 & 7^n-3^n is divisible by 4 .

Using the principle of mathematical induction, prove that 1.2+2.3+3.4+......+n(n+1)=(1)/(3)n(n+1)(n+2)

Using the principle of mathematical induction, prove that 1/(1*2)+1/(2*3)+1/(3*4)+…+1/(n(n+1)) = n/((n+1)) .

Using the principle of mathematical induction prove that 1/(1. 2. 3)+1/(2. 3. 4)+1/(3. 4. 5)++1/(n(n+1)(n+2))=(n(n+3))/(4(n+1)(n+2) for all n in N