Question

For each of the following, solve the given recurrence relation. Show your workings. (i) aₙ =1/​ 3 aₙ−₁ for n>1,a₁ =7.

Answer 1

To solve the recurrence relation an = 1/3 an-1 with a1 = 7, iterative substitution is employed. A pattern emerges, leading to a general solution where an is equal to 7/3(n-1).

Step-by-step explanation:

The question asks us to solve the recurrence relation an = 1/3 an-1 for n>1, with the initial condition a1 = 7. To solve this, we can use iterative substitution to find a pattern for an.

1. 
   
3. Start with the initial condition: a1 = 7.
4. 
   
6. Substitute n = 2 into the recurrence relation: a2 = 1/3 a1 = 1/3 × 7 = 7/3.
7. 
   
9. Substitute n = 3: a3 = 1/3 a2 = 1/3 × 7/3 = (7/3)/3 = 7/9.

This pattern suggests that for any positive integer n, an = 7/3(n-1).