Write the first five terms of each geometric sequence. a_n = - 4a_(n-1), a₁ = 10

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Identify the first term of the geometric sequence, which is given as \(a_1 = 10\).

Recognize the recursive formula for the sequence: \(a_n = -4a_{n-1}\), meaning each term is obtained by multiplying the previous term by \(-4\).

Calculate the second term using the formula: \(a_2 = -4 \times a_1\).

Calculate the third term using the formula: \(a_3 = -4 \times a_2\).

Continue this process to find the fourth and fifth terms: \(a_4 = -4 \times a_3\) and \(a_5 = -4 \times a_4\).

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주요 개념

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Geometric Sequence Definition

A geometric sequence is a sequence 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 term a_n depends on the previous term a_(n-1) multiplied by -4, which means each term is generated by applying this rule starting from the initial term.

Finding Terms of a Sequence

To find terms of a sequence given a recursive formula and the first term, start with the initial term and repeatedly apply the formula to find subsequent terms. For example, multiply the first term by the common ratio to get the second term, then use the second term to find the third, and so on.

Write the first five terms of each geometric sequence. an = - 4a(n-1), a1 = 10