Problem 7 - College Algebra

Question

Write the first five terms of each geometric sequence. $$a_{n}=-5 a_{n-1}, \quad a_{1}=-6$$

Step-by-Step Solution

Answer

The first five terms of the geometric sequence are -6, 30, -150, 750, -3750.

Step 1: Identify the First Term and the Common Ratio

The sequence is defined with the first term \(a_{1}=-6\), and common ratio -5 from the formula \(a_{n}=-5a_{n-1}\).

Step 2: Apply the Sequence Rule for the Second Term (\(a_{2}\))

To find the second term, insert 2 as \(n\) in the formula and use the value of \(a_{1}\). This gives \(a_{2}=-5 * a_{1} = -5 * -6 = 30\).

Step 3: Apply the Sequence Rule for the Third Term (\(a_{3}\))

Now we have \(a_{2}\), so we apply the formula again: \(a_{3}=-5 * a_{2} = -5 * 30 = -150\).

Step 4: Apply the Sequence Rule for the Fourth Term (\(a_{4}\))

Applying the formula again, we get \(a_{4}=-5 * a_{3} = -5 * -150 = 750\).

Step 5: Apply the Sequence Rule for the Fifth Term (\(a_{5}\))

We repeat the same process for the fifth term: \(a_{5}=-5 * a_{4} = -5 * 750 = -3750\).