Find a2 and a3 for each geometric sequence. 2, a2, a3, - 54

Ch. 8 - Sequences, Induction, and Probability

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Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

Problema 64

Capitolo 9, Problema 64

Find a₂ and a₃ for each geometric sequence. 2, a₂, a₃, - 54

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Identify the terms of the geometric sequence given: the first term $a_1 = 2$, the second term $a_2$, the third term $a_3$, and the fourth term $a_4 = -54$.

Recall that in a geometric sequence, each term is found by multiplying the previous term by a common ratio $r$. So, $a_2 = a_1 \times r$, $a_3 = a_2 \times r = a_1 \times r^2$, and $a_4 = a_1 \times r^3$.

Use the known value of the fourth term to set up an equation: $a_4 = a_1 \times r^3 = -54$. Substitute $a_1 = 2$ to get $2 \times r^3 = -54$.

Solve the equation $2 \times r^3 = -54$ for $r^3$ by dividing both sides by 2, then find $r$ by taking the cube root of both sides.

Once you have the value of $r$, calculate $a_2$ using $a_2 = 2 \times r$ and $a_3$ using $a_3 = 2 \times r^2$.

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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. Understanding this definition helps identify the relationship between consecutive terms.

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Geometric Sequences - Recursive Formula

Common Ratio Calculation

The common ratio (r) is found by dividing any term by its preceding term. Knowing how to calculate r allows you to find unknown terms in the sequence by multiplying the previous term by r.

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Graphs of Common Functions

Term Formula for Geometric Sequences

The nth term of a geometric sequence is given by a_n = a_1 * r^(n-1), where a_1 is the first term and r is the common ratio. This formula is essential for finding specific terms like a2 and a3 when some terms are known.

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Geometric Sequences - Recursive Formula

Pratica correlata

Domanda del libro di testo

In Exercises 61–68, use the graphs of and to find each indicated sum.

$\sum_{i = 1}^{5} (a_{i} + 3 b_{i})$