/*! 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 32 Find \(a_{1}\) for each arithmet... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 32

Find \(a_{1}\) for each arithmetic sequence. $$S_{28}=2926, a_{28}=199$$

Short Answer

Expert verified
The first term, \(a_1\), is 10.

Step by step solution

01

Understand the Formula for the Sum of an Arithmetic Sequence

The sum of the first \(n\) terms of an arithmetic sequence is given by the formula \( S_n = \frac{n}{2} (a_1 + a_n) \), where \( S_n \) is the sum of the first \(n\) terms, \( a_1 \) is the first term, and \( a_n \) is the nth term.
02

Substitute Known Values into the Sum Formula

Substitute \( n = 28 \), \( S_{28} = 2926 \), and \( a_{28} = 199 \) into the formula: \[ 2926 = \frac{28}{2} (a_1 + 199) \].
03

Simplify the Equation

Simplify the equation \( 2926 = 14(a_1 + 199) \) by dividing both sides by 14 to isolate \(a_1 + 199\): \[ 209 = a_1 + 199 \].
04

Solve for \(a_1\)

Subtract 199 from both sides of the equation \( 209 = a_1 + 199 \) to find \(a_1\): \[ 209 - 199 = a_1 \].
05

Final Calculation

Perform the subtraction: \( a_1 = 10 \). Therefore, the first term \(a_1\) is 10.

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 an Arithmetic Sequence
Understanding the sum of an arithmetic sequence is fundamental when working with this particular type of sequence. An arithmetic sequence is a list of numbers with a constant difference between consecutive terms. The sum of the first \( n \) terms of an arithmetic sequence is calculated using the formula:
  • \( S_n = \frac{n}{2} (a_1 + a_n) \)
where:
  • \( S_n \) represents the sum of the first \( n \) terms.
  • \( a_1 \) is the first term.
  • \( a_n \) is the nth term.

For the exercise in question, you have parameters: \( n = 28 \), \( S_{28} = 2926 \), and \( a_{28} = 199 \). By substituting these values into the formula, you can solve for the missing term. By understanding how to manipulate and utilize this formula, you can tackle various problems involving arithmetic sequences efficiently.
Finding the First Term
In many arithmetic sequence problems, finding the first term is a key step to understand the sequence as a whole. For instance, you may be given the sum of the sequence and an nth term, such as in the exercise: \[ 2926 = \frac{28}{2} (a_1 + 199) \] Simplifying this is typically the first step. The equation can be reworked by dividing both sides by 14, leading to:\[ 209 = a_1 + 199 \]To isolate \( a_1 \), subtract the given nth term (199) from both sides. This gives:\[ a_1 = 209 - 199 \]Thus, \( a_1 = 10 \). By following these steps methodically, you can determine the first term of the sequence, which is vital for further calculations and understanding the sequence's pattern.
Sequence Formula
To effectively work with any sequence, especially arithmetic sequences, being familiar with the sequence formula is crucial. Arithmetic sequence formulas help identify properties of the sequence and solve for unknowns. The general formula for the \( n \)-th term of an arithmetic sequence is:\[ a_n = a_1 + (n-1)d \]where:
  • \( a_n \) is the nth term.
  • \( a_1 \) is the first term.
  • \( d \) is the common difference between the terms.

Although in the exercise's solution, the term \( a_{28} \) is already known, understanding the formula is critical to solving for the sequence further should additional information be required or if \( d \) needs to be identified. It allows any sequence to be analyzed or extended as needed dependably.

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 the positive integer values of \(n\) for which the given statement is not true. $$3^{n}>6 n$$

MODELING Disease Infection What will happen when an infectious disease is introduced into a family? Suppose a family has \(I\) infected members and \(S\) members who are not infected, but are susceptible to contracting the disease. The probability \(P\) of \(k\) people not contracting the disease during a l-week period can be calculated by the formula $$P=\left(\begin{array}{l} S \\ k \end{array}\right) q^{k}(1-q)^{s-k}$$ where \(q=(1-p)^{l}\) and \(p\) is the probability that a susceptible person contracts the disease from an infected person. For example, if \(p=0.5,\) then there is a \(50 \%\) chance that a susceptible person exposed to one infected person for 1 week will contract the disease. (Source: Hoppensteadt, F. and C. Peskin, Mathematics in Medicine and the Life Sciences, Springer-Verlag.) A. Approximate the probability \(P\) of 3 family members not becoming infected within 1 week if there are currently 2 infected and 4 susceptible members. Assume that \(p=0.1\) B. A highly infectious disease can have \(p=0.5 .\) Repeat part (a) with this value of \(p\) C. Approximate the probability that everyone would become sick in a large family if initially \(I=1, S=9\) and \(p=0.5\)

Use any or all of the methods described in this section to solve each problem. In an experiment on plant hardiness, a researcher gathers 6 wheat plants, 3 barley plants, and 2 rye plants. She wishes to select 4 plants at random. (a) In how many ways can this be done? (b) In how many ways can it be done if exactly 2 wheat plants must be included?

Gender Makeup of a Family Suppose a family has 5 children. Suppose also that the probability of having a girl is \(\frac{1}{2} .\) Find the probability that the family has the following children. (picture cannot copy) At least 3 boys

Use a formula to find the sum of each arithmetic series. $$7.5+6+4.5+3+1.5+0+(-1.5)$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Geometry

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.