Chapter 12.1, Problem 82E

To determine

Calculate the first $10$ terms of the sequence. Find whether the recursive sequence different from the others in this section.

Answer to Problem 82E

The first ten terms of sequence are : $1,1,2,3,3,6,4,4,6,9$ .In the sequence it can be seen that the first two terms of sequence are same but the third term are different and again next two term (fourth and fifth) are same but the sixth term are different and again next two term (seventh and eighth) are same and this way the sequence goes.

Explanation of Solution

Given: It is given in the question that the sequence is $a_n=a_{n-a_{n-1}}+a_{n-a_{n-2}}$ with $a_1=1and{a}_2=1$ .

Concept Used:

In this, use the concept of putting the several value in the given sequence and get the term.

Calculation:

Here, the sequence is $a_n=a_{n-a_{n-1}}+a_{n-a_{n-2}}$ with $a_1=1and{a}_2=1$ .

  $\begin{array}{l}{a}_3=a_{3-a_2}+a_{3-a_1}=a_2+a_2=1+1=2\\ a_4=a_{4-a_3}+a_{4-a_2}=a_2+a_3=1+2=3\\ a_5=a_{5-a_4}+a_{5-a_3}=a_2+a_3=1+2=3\\ a_6=a_{6-a_5}+a_{6-a_4}=a_3+a_3=3+3=6\\ a_7=a_{7-a_6}+a_{7-a_5}=a_1+a_2=3+1=4\\ a_8=a_{8-a_7}+a_{8-a_6}=a_4+a_2=3+1=4\\ a_9=a_{9-a_8}+a_{9-a_7}=a_4+a_5=3+3=6\\ a_{10}=a_{10-a_9}+a_{10-a_8}=a_4+a_6=3+6=9\end{array}$

Hence, the first ten terms of sequence are : $1,1,2,3,3,6,4,4,6,9$ .

Here, in the above sequence it can be seen that the first two terms of sequence are same but the third term are different and again next two term (fourth and fifth) are same but the sixth term are different and again next two term (seventh and eighth) are same and this way the sequence goes.

Conclusion:

The first ten terms of sequence are : $1,1,2,3,3,6,4,4,6,9$ .