Math Problem Statement

The nth term of a sequence is given: a_n = 5(-0.5)^(n-1). (a) Find the first five terms of the sequence. (b) What is the common ratio r?

Solution

Let's solve the problem step by step based on the given formula for the sequence:

The formula for the nn-th term of the sequence is: an=5(0.5)n1a_n = 5(-0.5)^{n-1}


(a) Find the first five terms:

We calculate a1,a2,a3,a4,a_1, a_2, a_3, a_4, and a5a_5 by substituting n=1,2,3,4,5n = 1, 2, 3, 4, 5 into the formula.

  1. a1=5(0.5)11=5(0.5)0=5(1)=5a_1 = 5(-0.5)^{1-1} = 5(-0.5)^0 = 5(1) = 5
  2. a2=5(0.5)21=5(0.5)1=5(0.5)=2.5a_2 = 5(-0.5)^{2-1} = 5(-0.5)^1 = 5(-0.5) = -2.5
  3. a3=5(0.5)31=5(0.5)2=5(0.25)=1.25a_3 = 5(-0.5)^{3-1} = 5(-0.5)^2 = 5(0.25) = 1.25
  4. a4=5(0.5)41=5(0.5)3=5(0.125)=0.625a_4 = 5(-0.5)^{4-1} = 5(-0.5)^3 = 5(-0.125) = -0.625
  5. a5=5(0.5)51=5(0.5)4=5(0.0625)=0.3125a_5 = 5(-0.5)^{5-1} = 5(-0.5)^4 = 5(0.0625) = 0.3125

Thus, the first five terms are: a1=5,a2=2.5,a3=1.25,a4=0.625,a5=0.3125a_1 = 5, \, a_2 = -2.5, \, a_3 = 1.25, \, a_4 = -0.625, \, a_5 = 0.3125


(b) Find the common ratio rr:

The common ratio rr in a geometric sequence is the ratio between consecutive terms: r=an+1anr = \frac{a_{n+1}}{a_n}

Using a1=5a_1 = 5 and a2=2.5a_2 = -2.5: r=a2a1=2.55=0.5r = \frac{a_2}{a_1} = \frac{-2.5}{5} = -0.5

Thus, the common ratio is: r=0.5r = -0.5


Final Answer:

(a) First five terms:
a1=5,a2=2.5,a3=1.25,a4=0.625,a5=0.3125a_1 = 5, \, a_2 = -2.5, \, a_3 = 1.25, \, a_4 = -0.625, \, a_5 = 0.3125

(b) Common ratio:
r=0.5r = -0.5


Let me know if you'd like a detailed explanation of any part! 😊
Here are five related questions to deepen your understanding:

  1. What is the general formula for the sum of the first nn terms of a geometric sequence?
  2. How does the sign of rr affect the terms of the sequence?
  3. If r<1|r| < 1, what happens to the sequence as nn \to \infty?
  4. What would a10a_{10} and a15a_{15} be for this sequence?
  5. How would the sequence change if r=+0.5r = +0.5 instead of 0.5-0.5?

Tip: Always confirm the common ratio by calculating the ratio for multiple consecutive terms to ensure consistency in a geometric sequence.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Sequences
Geometric Sequence

Formulas

nth term of a geometric sequence: a_n = a * r^(n-1)
Common ratio: r = a_(n+1) / a_n

Theorems

Properties of geometric sequences

Suitable Grade Level

Grades 9-11