/*! 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 21 Find \(a_{8}\) and \(a_{n}\) for... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 21

Find \(a_{8}\) and \(a_{n}\) for each arithmetic sequence. $$a_{1}=x, a_{2}=x+3$$

Short Answer

Expert verified
The eighth term is \(a_8 = x + 21\) and the general term is \(a_n = 3n - 3 + x\).

Step by step solution

01

Identify the Common Difference

The given arithmetic sequence starts with \(a_1 = x\) and \(a_2 = x + 3\). To find the common difference \(d\), subtract \(a_1\) from \(a_2\): \(d = (x + 3) - x = 3\).
02

Use the Formula for the General Term

The formula for the \(n\)-th term in an arithmetic sequence is \(a_n = a_1 + (n-1)\cdot d\). Here, \(a_1 = x\) and \(d = 3\). Substitute these values into the formula: \(a_n = x + (n-1)\cdot 3\).
03

Calculate the Eighth Term

To find the eighth term, substitute \(n = 8\) into the formula: \(a_8 = x + (8-1)\cdot 3 = x + 21\). So, \(a_8 = x + 21\).
04

Express the General Term for the Sequence

Based on Step 2, the general term for the sequence is \(a_n = x + 3(n-1)\), which simplifies to \(a_n = 3n - 3 + x\).

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.

Understanding Common Difference
An arithmetic sequence is a series of numbers with a consistent difference between consecutive terms. This difference is known as the 'common difference'. To identify this, subtract the first term from the second. In our example, the common difference \(d\) is calculated by taking the difference between the second term \(a_2 = x + 3\) and the first term \(a_1 = x\). The resulting calculation, \(d = (x + 3) - x\), simplifies to \(d = 3\). This shows that each term in the sequence increases by 3 compared to the previous term.
  • The common difference is what keeps the sequence uniform.
  • It's the key to figuring out future terms in the sequence.
  • Once determined, solving for any term gets straightforward.
Exploring the n-th Term Formula
The n-th term formula is a powerful tool in any arithmetic sequence. This formula provides a way to find any term in the sequence, given its position \(n\). The general form of the n-th term formula in an arithmetic sequence is \(a_n = a_1 + (n-1) \cdot d\).

First, identify your initial term, \(a_1\), and your common difference, \(d\). For our sequence, \(a_1 = x\) and \(d = 3\). Substituting these into the formula, we get \(a_n = x + (n-1) \cdot 3\), providing a blueprint to compute any term's value in the sequence. It's important as it enables direct calculation of terms without having to write out the entire sequence.
Determining the General Term
The general term of an arithmetic sequence describes any term's position in terms of a formula. The general term formula for our exercise, based on the given information and n-th term formula, becomes \(a_n = 3n - 3 + x\). This form represents every term in the sequence by expressing a relationship between the position \(n\) and the corresponding term value.

Key points to remember:
  • The general term is crucial for expressing any term in the sequence compactly.
  • It allows you to find specific terms quickly, such as the 8th term, without excessive calculations.
  • This efficiency is important for working with large sequences or when solving problems that require term identification.
Knowing the general term offers a deeper understanding of the sequence's structure and behavior, simplifying analysis and problem-solving.

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

Prove each statement for positive integers \(n\) and \(r\), with \(r \leq n\). (Hint: Use the definitions of permutations and combinations.) $$P(n, n-1)=P(n, n)$$

Prove each statement by mathematical induction. \(2^{n}>n^{2},\) for \(n>4\)

Solve each problem. Number of Handshakes Suppose that each of the \(n\) \((n \geq 2)\) people in a room shakes hands with everyone else, but not with himself. Show that the number of handshakes is \(\frac{n^{2}-n}{2}\)

Prove each statement for positive integers \(n\) and \(r\), with \(r \leq n\). (Hint: Use the definitions of permutations and combinations.) $$C(n, 0)=1$$

Find all arithmetic sequences \(a_{1}, a_{2}\) \(a_{3}, \ldots\) such that \(a_{1}^{2}, a_{2}^{2}, a_{3}^{2}, \ldots\) is also an arithmetic sequence.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Statistics

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.