Question

DISCUSS: A Different Type of Recursion Find the first ten terms of the sequence defined by $a_n=a_{n-a_{n-1}}+a_{n-a_{n-2}}$with $a_1=1\text{and}{a}_2=1$

How is this recursive sequence different from the others in this section?

Short answer

The first ten step of the $a_n=a_{n-a_{n-1}}+a_{n-a_{n-1}}$ will be $\text{1,1,2,2,2,2,2,2,2,2}..$

Step-by-step solution

Step 1.Given information.

Given: $a_1=1\text{and}{a}_2=1$

Step 2. Find the first ten terms.

Use the given formula.

First of all, the sequence is recursive. for finding step${\text{a}}_3$we need to find$a_{n-1}$

Here,$a_2=1$that's why:

For n=3,

$\begin{array}{l}{a}_{\left\{3\right\}}=a_{\{3-a_3\}}+a_{\{3-a_2\}}\\ =a_{\left\{2\right\}}+a_{\left\{2\right\}}\\ =1+1=2\end{array}$

For n=4,

$\begin{array}{l}{a}_{\left\{4\right\}}=a_{\left\{4-a_3\right\}}+a_{\left\{4-a_2\right\}}\\ =a_{\left\{2\right\}}+a_{\left\{3\right\}}\\ =1+2\\ =3\end{array}$

For n=5,

$\begin{array}{l}{a}_{\left\{5\right\}}=a_{\left\{5-a_4\right\}}+a_{\left\{5-a_3\right\}}\\ =a_{\left\{2\right\}}+a_{\left\{3\right\}}\\ =1+2\\ =3\end{array}$

For n=6,

$\begin{array}{l}{a}_{\left\{6\right\}}=a_{\left\{6-a_5\right\}}+a_{\left\{6-a_4\right\}}\\ =a_{\left\{3\right\}}+a_{\left\{3\right\}}\\ =3+3\\ =6\end{array}$

For n=7,

$\begin{array}{l}{a}_{\left\{7\right\}}=a_{\left\{7-a_6\right\}}+a_{\left\{7-a_5\right\}}\\ =a_{\left\{1\right\}}+a_{\left\{4\right\}}\\ =1+3\\ =4\end{array}$

For n=8,

$\begin{array}{l}{a}_{\left\{8\right\}}=a_{\left\{8-a_7\right\}}+a_{\left\{8-a_6\right\}}\\ =a_{\left\{4\right\}}+a_{\left\{2\right\}}\\ =3+1=4\end{array}$

For n=9,

$\begin{array}{l}{a}_{\left\{9\right\}}=a_{\left\{9-a_8\right\}}+a_{\left\{9-a_7\right\}}\\ =a_{\left\{5\right\}}+a_{\left\{5\right\}}\\ =3+3\\ =6\end{array}$

For n=10,

$\begin{array}{l}{a}_{\left\{10\right\}}=a_{\left\{10-a_9\right\}}+a_{\left\{10-a_8\right\}}\\ =a_{\left\{4\right\}}+a_{\left\{6\right\}}\\ =3+6\\ =9\end{array}$

Hence, first ten terms of the sequence are $\text{1,1,2,3,3,6,4,4,6,9}$

If we talk about a burst of 3 digits then first 2 terms of each burst are the same but the third digit is different,
1st,2ndis the same but3rdis different, 4th,5th is the same but 6th is different.
This pattern is followed in the entire sequence.
Hence this pattern differentiate this sequence from others give in this section.