Question

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

$a_n=\frac{a_{n-1}}{6}\text{and}{a}_1=-24$

Short answer

The first five terms are $-24, -4, -\frac{2}{3}, -\frac{1}{9}, -\frac{1}{54}$.

Step-by-step solution

Step 1.  Given

The given sequence is $a_n=\frac{a_{n-1}}{6} \text{and} a_1=-24$.

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.

$\begin{array}{l}\text{For\hspace{0.17em}\hspace{0.17em}}n=1,a_1=-24\\ \text{For\hspace{0.17em}\hspace{0.17em}}n=2,a_2=\frac{a_{2-1}}{6}=\frac{a_1}{6}=\frac{-24}{6}=-4\\ \text{For\hspace{0.17em}\hspace{0.17em}}n=3,a_3=\frac{a_{3-1}}{6}=\frac{a_2}{6}=\frac{-4}{6}=-\frac{2}{3}\\ \text{For\hspace{0.17em}\hspace{0.17em}}n=4,a_4=\frac{a_{4-1}}{6}=\frac{a_3}{6}=\frac{-\frac{2}{3}}{6}=-\frac{2}{18}=-\frac{1}{9}\\ \text{For\hspace{0.17em}\hspace{0.17em}}n=5,a_5=\frac{a_{5-1}}{6}=\frac{a_4}{6}=\frac{-\frac{1}{9}}{6}=-\frac{1}{54}\end{array}$

Hence, the first five terms are$-24,-4,-\frac{2}{3},-\frac{1}{9},-\frac{1}{54}$