/*! 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 8 For a geometric sequence with fi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 8

For a geometric sequence with first term \(a_{1}\) and common ratio \(r,\) where \(r \neq 0, r \neq 1,\) the sum of the first \(n\) terms is \(S_{n}=a_{1} \cdot \frac{1-r^{n}}{1-r}\)

Short Answer

Expert verified
Use \(S_n = a_1 \frac{1 - r^n}{1 - r}\) to find the sum of the first \(n\) terms of a geometric sequence.

Step by step solution

01

Identify the given terms

A geometric sequence is defined by its first term, denoted as \(a_1\), and its common ratio, denoted as \(r\). Given: \(r eq 0\) and \(r eq 1\).
02

Understand the formula for the sum of the first n terms

The sum of the first \(n\) terms of a geometric sequence \(S_n\) is given by:\[ S_n = a_1 \frac{1 - r^n}{1 - r} \]This formula accounts for the series where \(r\) is not equal to 1 or 0.
03

Break down the formula components

\(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms. The numerator \(1 - r^n\) captures the effect of the ratio over \(n\) terms, and the denominator \(1 - r\) normalizes the sum.
04

Substitute values (if given)

If specific values for \(a_1\), \(r\), and \(n\) are provided, substitute them into the formula to find the sum \(S_n\).
05

Simplify the expression

After substitution, simplify the expression to find the sum.

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.

sum of series
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. We often need to find the sum of a given number of terms in such a sequence. The sum of the first n terms of a geometric series is given by the formula:
\[ S_n = a_1 \frac{1 - r^n}{1 - r} \]
Here, \(S_n\) is the sum of the first n terms, \(a_1\) is the first term, and \(r\) is the common ratio. This formula helps us quickly compute the sum without having to add each term individually.
common ratio
The common ratio, denoted as \(r\), is a crucial part of a geometric sequence. It is the factor by which we multiply each term to get the next term in the sequence. For example, in the sequence 2, 6, 18, 54, ... the common ratio is 3 because each term is 3 times the previous one.
The common ratio must not be 0 or 1:
  • If \(r = 0\), the sequence would just be zeros after the first term.
  • If \(r = 1\), the sequence would just repeat the first term.
Understanding the common ratio helps in using the formula for the sum of the terms effectively.
first term
The first term of a geometric sequence, typically denoted as \(a_1\), plays a significant role in determining the sequence's behavior. It is the starting point of the sequence. For example, in the sequence 5, 10, 20, 40, ... the first term \(a_1\) is 5.
Knowing the first term is essential because it is used directly in the sum formula. The sum of the first n terms depends on this term, as it is multiplied by a factor to account for the common ratio and the number of terms.
formula derivation
Deriving the formula for the sum of the first n terms of a geometric sequence involves a few steps. Here's how you can understand it:
  • Start with the geometric sequence: \(a_1, a_1r, a_1r^2, ..., a_1r^{n-1}\).
  • Write the sum of these terms: \(S_n = a_1 + a_1r + a_1r^2 + ... + a_1r^{n-1}\).
  • To derive the formula, multiply both sides of the equation by the common ratio \(r\):
    \(rS_n = a_1r + a_1r^2 + a_1r^3 + ... + a_1r^n\).
  • Subtract the second equation from the first:
    \(S_n - rS_n = a_1 - a_1r^n\).
  • Factor out \(S_n\) and \(a_1\):
    \(S_n(1 - r) = a_1(1 - r^n)\).
  • Solve for \(S_n\):
    \[ S_n = a_1 \frac{1 - r^n}{1 - r} \]
And that's the derived formula to calculate the sum of the first n terms in a geometric sequence!

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

Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. If the sequence is arithmetic or geometric, find the sum of the first 50 terms. $$ 2,4,6,8, \ldots $$

Bode's Law In \(1772,\) Johann Bode published the following formula for predicting the mean distances, in astronomical units (AU), of the planets from the sun: $$ a_{1}=0.4 \quad a_{n}=0.4+0.3 \cdot 2^{n-2} $$ where \(n \geq 2\) is the number of the planet from the sun. (a) Determine the first eight terms of the sequence. (b) At the time of Bode's publication, the known planets were Mercury \((0.39 \mathrm{AU}),\) Venus \((0.72 \mathrm{AU}),\) Earth \((1 \mathrm{AU})\) Mars \((1.52 \mathrm{AU}),\) Jupiter \((5.20 \mathrm{AU}),\) and Saturn \((9.54 \mathrm{AU})\) How do the actual distances compare to the terms of the sequence? (c) The planet Uranus was discovered in \(1781,\) and the asteroid Ceres was discovered in \(1801 .\) The mean orbital distances from the sun to Uranus and Ceres " are \(19.2 \mathrm{AU}\) and \(2.77 \mathrm{AU},\) respectively. How well do these values fit within the sequence? (d) Determine the ninth and tenth terms of Bode's sequence. (e) The planets Neptune and Pluto" were discovered in 1846 and \(1930,\) respectively. Their mean orbital distances from the sun are \(30.07 \mathrm{AU}\) and \(39.44 \mathrm{AU},\) respectively. How do these actual distances compare to the terms of the sequence? (f) On July \(29,2005,\) NASA announced the discovery of a dwarf planet \((n=11),\) which has been named Eris. Use Bode's Law to predict the mean orbital distance of Eris from the sun. Its actual mean distance is not yet known, but Eris is currently about 97 astronomical units from the sun.

How many terms must be added in an arithmetic sequence whose first term is 11 and whose common difference is 3 to obtain a sum of \(1092 ?\)

Expand each expression using the Binomial Theorem. $$ \left(x^{2}-y^{2}\right)^{6} $$

Don contributes \(\$ 500\) at the end of each quarter to a tax-sheltered annuity (TSA). What will the value of the TSA be after the 80 th deposit ( 20 years) if the per annum rate of return is assumed to be \(5 \%\) compounded quarterly?
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

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.