Question

15-20. Recursive Sequences A sequence is defined recursively by the given formulas. Find the first five terms of the sequence.

20. $a_n=a_{n-1}+a_{n-2}+a_{n-3}\text{and}{a}_1=a_2=a_3=1$.

Short answer

The first five terms are 1, 1, 1, 3, 5.

Step-by-step solution

Step-1 – Given

The given sequence is$a_n=a_{n-1}+a_{n-2}+a_{n-3} \text{and} a_1=a_2=a_3=1$.

Step-2 – To determine

We have to find the first five terms of the given sequence.

Step-3 – Calculation

To find the first five terms of the sequence we’ll plug n = 1, 2, 3, 4 and 5 respectively in the given sequence.

$\begin{array}{l}\text{For}n=1,a_1=1\\ \text{For}n=2,a_2=1\\ \text{For}n=3,a_3=1\\ \text{For}n=4,a_4=a_{4-1}+a_{4-2}+a_{4-3}=a_3+a_2+a_1=1+1+1=3\\ \text{For}n=5,a_5=a_{5-1}+a_{5-2}+a_{5-3}=a_4+a_3+a_2=3+1+1=5\end{array}$

Hence, the first five terms are 1, 1, 1, 3, 5.