Chapter 8, Problem 7CR

To determine

The first five terms of the given recursive sequence.

Answer to Problem 7CR

The first five terms of the given recursive sequence are-

9, 3, 1, $\frac{1}{3}$ and $\frac{1}{9}$ .

Explanation of Solution

Given: The given recursive sequence is defined as-

  $a_1=9$ ; $a_{k+1}=\frac{1}{3}{a}_k$

Concept Used: A recursive sequence is a type of sequence in with the values of previous terms are used to find the value of a succeeding term. Hence, the value of $a_{n-1}$ should be known to find the value of $a_n$ . At times more than one previous terms might be needed to find the value of a succeeding term. Generally, the $n^{th}$ term of a recursive sequence consists of the value of a known term and a recursive formula that depends on the known term, which is used to find the value of all other terms in the sequence.

Calculations: The given recursive sequence is defined as-

  $a_1=9$ ; $a_{k+1}=\frac{1}{3}{a}_k$

The first term is 9. Putting the value of k = 1, 2, 3 … in the given recursive formula, the other terms of the sequence are-

  $a_1=9$

  $a_2=\frac{1}{3}{a}_1=\frac{1}{3}\times 9=3$ (Putting k = 1 in the recursive formula)

  $a_3=\frac{1}{3}{a}_2=\frac{1}{3}\times 3=1$ (Putting k = 2 in the recursive formula)

  $a_4=\frac{1}{3}{a}_3=\frac{1}{3}\times 1=\frac{1}{3}$ (Putting k = 3 in the recursive formula)

  $a_5=\frac{1}{3}{a}_4=\frac{1}{3}\times \frac{1}{3}=\frac{1}{9}$ (Putting k = 4 in the recursive formula)

Hence, the first five terms of the sequence are-

9, 3, 1, $\frac{1}{3}$ and $\frac{1}{9}$ .