/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 43 Find \(S_{n}\) for each geometri... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 43

Find \(S_{n}\) for each geometric series described. $$ a_{3}=-36, a_{6}=-972, n=10 $$

Short Answer

Expert verified
The sum of the first 10 terms is -118096.

Step by step solution

01

Identify the common ratio (r)

The formula for the nth term of a geometric sequence is given by \(a_n = a_1 imes r^{n-1}\). We know \(a_3 = -36\) and \(a_6 = -972\). Let鈥檚 express both terms in terms of \(a_1\) and \(r\):\[-36 = a_1 imes r^2\] and \[-972 = a_1 imes r^5.\] By dividing the second equation by the first, we can solve for \(r\):\[\frac{-972}{-36} = \frac{a_1 imes r^5}{a_1 imes r^2} \27 = r^3 \] Taking the cube root from both sides, \[r = 3.\]
02

Find the first term (a_1)

Now that we have the common ratio, substitute \(r = 3\) back into either equation to find \(a_1\). Using \(-36 = a_1 imes 3^2\):\[a_1 imes 9 = -36\a_1 = \frac{-36}{9} = -4.\]
03

Calculate the sum of the first n terms

The formula for the sum of a geometric series is \[ S_{n} = a_1 \times \frac{1-r^n}{1-r} \] where \(a_1 = -4\), \(r=3\), and \(n=10\). Substitute these values into the formula:\[S_{10} = -4 \times \frac{1-3^{10}}{1-3}. \] Calculate \(3^{10} = 59049\):\[S_{10} = -4 \times \frac{1-59049}{-2} \S_{10} = -4 \times 29524 \S_{10} = -118096.\]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Ratio
A geometric series is identified by its constant factor between consecutive terms, known as the common ratio. In a geometric sequence, each term after the first is obtained by multiplying the previous one by this fixed value, often denoted as \( r \). To find \( r \), you can utilize the formula for two known terms: \[ a_3 = a_1 \times r^2 \] and \[ a_6 = a_1 \times r^5. \]
Divide these equations to eliminate \( a_1 \) and solve for \( r \): \[ \frac{-972}{-36} = r^3 \] and therefore, \( r = 3 \). Remember:
  • \( r=1 \) leads to a constant series.
  • Greater values of \( r \) can result in rapidly increasing or decreasing sequences.
Nth Term
The nth term formula in a geometric sequence tells us the value at a particular position. It is represented as \( a_n = a_1 \times r^{n-1} \). This formula helps compute any specific term if the first term and the common ratio are known.
For example, if \( a_1 = -4 \) and \( r = 3 \), the formula can find any term, like the sixth term \( a_6 \). By substituting into the formula: \[ a_6 = -4 \times 3^{5} = -972. \] Note that:
  • The term \( a_1 \) is the initial value of the sequence.
  • The exponent \( n-1 \) in \( r^{n-1} \) indicates how many times the ratio is applied.
Series Sum
The sum of the terms in a geometric series can be essential in scenarios like calculating interest over years or understanding growth patterns. To find the sum of the first \( n \) terms, use the formula:
\[ S_n = a_1 \times \frac{1-r^n}{1-r}. \]
In this problem, where \( a_1 = -4 \), \( r = 3 \), and \( n = 10 \):
  • Compute \( r^n \), which is \( 3^{10} = 59049. \)
  • Substitute into the formula:
    \[ S_{10} = -4 \times \frac{1-59049}{1-3} = -4 \times 29524 = -118096. \]
The negative value here implies a sum moving in a downward trajectory over these ten terms.
First Term
The first term of a geometric series, denoted as \( a_1 \), serves as the starting point for the entire sequence. It gets transformed by the common ratio \( r \) to yield subsequent terms.
To find \( a_1 \), you can rearrange the nth term formula. With \( a_3 = -36 \) and \( r = 3 \), the expression becomes:
\[ -36 = a_1 \times 9 \] so \[ a_1 = \frac{-36}{9} = -4. \] This parameter is crucial because:
  • It directly influences the position and behavior of the sequence.
  • Changes in \( a_1 \) will alter the entire series' trajectory and values.
Understanding \( a_1 \) helps map out the evolution of the entire series from its inception.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Evaluate each expression. $$ \frac{9 !}{7 !} $$

OPEN ENDED Write a power of a binomial for which the first term of the expansion is 625\(x^{4}\) .

Prove that each statement is true for all positive integers. \(5^{n}+3\) is divisible by 4

Expand each power. $$ \left(3+\frac{m}{3}\right)^{5} $$

State whether each statement is true or false when \(n=1\). Explain. $$ 1=\frac{n(n+1)}{2} $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.