Problem 6

Write the first five terms of each geometric sequence. $$a_{n}=-3 a_{n-1}, \quad a_{1}=10$$

Verified

Answer

The first five terms of the geometric sequence are: 10, -30, 90, -270, 810.

1Step 1: Identifying the First Term

As given in the question, the first term (also called the initial term or $a_1$) of the geometric sequence is 10.

2Step 2: Using the recursive formula to find the second term

Use the recursive formula $a_{n}=-3 a_{n-1}$ by substituting $n=2$ and $a_{1}=10$ to find the second term $a_2$: $a_2=-3*a_1=-3*10=-30$

3Step 3: Calculating the third term

Again use the recursive formula $a_{n}=-3 a_{n-1}$ by inputting $n=3$ and $a_{2}=-30$ to find the third term $a_3$: $a_3=-3*a_2=-3*(-30)=90$

4Step 4: Finding the fourth term

Repeating the previous step with $n=4$ and $a_{3}=90$ will provide the fourth term $a_4$: $a_4=-3*a_3=-3*90=-270$

5Step 5: Determining the fifth term

Finally, use the formula once more with $n=5$ and $a_{4}=-270$ to achieve the fifth term $a_5$: $a_5=-3*a_4=-3*(-270)=810$