In Exercises 1–8, write the first five terms of each geometric sequence. a_n = - 4a_(n-1), a₁ = 10

Verified step by step guidance

1. Identify the first term of the geometric sequence, which is given as $a{1}_{}=10$.
2. Recognize that the recursive formula is $a{n}_{}=\; -4a{\mathrm{n-1}}_{}$, meaning each term is obtained by multiplying the previous term by $-4$.
3. Calculate the second term by multiplying the first term by $-4$: $a{2}_{}=\; -4\; imes\; 10$.
4. Find the third term by multiplying the second term by $-4$: $a{3}_{}=\; -4\; imes\; a\_2$.
5. Continue this process to find the fourth and fifth terms by multiplying the previous term by $-4$ each time.

Key Concepts

Geometric Sequence Definition

A geometric sequence is a list of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. This ratio can be positive, negative, or fractional, and it determines the pattern of the sequence.

Recursive Formula for Sequences

A recursive formula defines each term of a sequence using one or more previous terms. In this problem, the formula an = -4a_(n-1) means each term is -4 times the previous term, which helps generate the sequence step-by-step starting from the initial term.

Finding Terms of a Sequence

To find terms of a sequence using a recursive formula, start with the given first term and repeatedly apply the formula to find subsequent terms. For example, with a1 = 10 and an = -4a_(n-1), calculate a2 by multiplying a1 by -4, then use a2 to find a3, and so on.