A sequence a0al a2 a3 is defined recursively as follows Use

A sequence a_0,a_l, a_2, a_3, ... is defined recursively as follows: Use the given recursion formula to evaluate a_n for n = 1,2,3,4. Use the results of (i) to guess a general formula for an. Use induction to prove that the formula from (ii) is correct.

Solution

(i) a_o = 1, For n=0, a₁ = 1/1+1 = 1/2

For n=1, a₂ = (1/2) / (1/2 +1) = 1/3

For n=2, a₃ = (1/3) / (1/3 +1) = 1/4

For n=3, a₄ = (1/4) / (1/4 +1) = 1/5

For n=4, a₅ = (1/5) / (1/5 +1) = 1/6

(ii) this forms the series,

     1, 1/2 , 1/3 , 1/4 , 1/5 , 1/6 , .......... 1/n+1

therefore   a_n = 1/n+1

(iii) a_n = 1/n+1,

now a_k = 1/k+1 it sould also be true for k+1,

a_k+1 = 1/k+2, this can be written as 1/(k+1)+1

for n=1., LHS a₁ = 1/2

RHS 1/n+1 = 1/2

Hence proved the formula by induction