Home
Class 12
MATHS
If a(1)=1,a(n+1)=(1)/(n+1)a(n),a ge1, th...

If `a_(1)=1,a_(n+1)=(1)/(n+1)a_(n),a ge1`, then prove by induction that `a_(n+1)=(1)/((n+1)!)n in N`.

Text Solution

Verified by Experts

Let `P(n):a_(n+1)=(1)/((n+1)!),n in N`.....(i)
where `a_1=1 and a_(n+1)=(1)/((n+1))a_(n),n ge 1` ......(ii)
Step I For n=1, form Eq. (i) , we get `a_(2)=(1)/((1+1)!)=(1)/(2!)`
But from Eq. (ii) , we get a_(2)=(1)/((1+1)),a_(1)=(1)/(2)(1)=(1)/(2)`
which is true ,
Also, for n=2 from Eq. (i) we get `a_3=(1)/(3!)=(1)/(6)`
But from Eq. (ii) , we get `a_3=(1)/(3),a_2=(1)/(3).(1)/(2)=(1)/(6)`
which is also true .
Hence ,P(1) and
Promotional Banner

Similar Questions

Explore conceptually related problems

Prove by induction that n(n+1) (2n+1) is divisible by 6.

Let a_(0)=2,a_1=5 and for n ge 2, a_n=5a_(n-1)-6a_(n-2) . Then prove by induction that a_(n)=2^(n)+3^(n) forall n in Z^+ .

If a_(1)=1, a_(2)=5 and a_(n+2)=5a_(n+1)-6a_(n), n ge 1 , show by using mathematical induction that a_(n)=3^(n)-2^(n)

If a_(1),a_(2),a_(3),".....",a_(n) are in HP, than prove that a_(1)a_(2)+a_(2)a_(3)+a_(3)a_(4)+"....."+a_(n-1)a_(n)=(n-1)a_(1)a_(n)

If a_(n) = (n^(2))/(2^(n)) , then find a_(7) .

If a_(1) , a_(2),"………",a_(n) are n non-zero real numbers such that ( a_(1)^(2) +a_(2)^(2) + "........."+a_(n-1)^(2) ) ( a_(2)^(2) + a_(3)^(2) + "........"+a_(n)^(2))le(a_(1) a_(2) + a_(2) a_(3) +".........." +a_(n-1) a_(n))^(2), a_(1), a_(2),".........",a_(n) are in :

If 1, a_(1), a_(2). ..., a_(n-1) are n roots of unity, then the value of (1 - a_(1)) (1 - a_(2)) .... (1 - a_(n-1)) is :

If a_(1),a_(2),a_(3)"....." are in GP with first term a and common ratio r, then (a_(1)a_(2))/(a_(1)^(2)-a_(2)^(2))+(a_(2)a_(3))/(a_(2)^(2)-a_(3)^(2))+(a_(3)a_(4))/(a_(3)^(2)-a_(4)^(2))+"....."+(a_(n-1)a_(n))/(a_(n-1)^(2)-a_(n)^(2)) is equal to

If a_(1)=2 and a_(n)=2a_(n-1)+5 for ngt1 , the value of sum_(r=2)^(5)a_(r) is

Find the sixth term of the sequence a_(n) =(n)/(n+1) ?