/*! 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 The nth term of a sequence is gi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 32

The nth term of a sequence is given. Find the first 4 terms; the 10 th term, \(a_{10} ;\) and the 15 th term, \(a_{15},\) of the sequence. $$ a_{n}=(-1)^{n} n^{2} $$

Short Answer

Expert verified
The first 4 terms are -1, 4, -9, and 16. a_{10} = 100 and a_{15} = -225.

Step by step solution

01

- Find the 1st term (n=1)

Substitute n = 1 into the given formula a_n = (-1)^n n^2. This gives us: a_1 = (-1)^1 * 1^2 = -1.
02

- Find the 2nd term (n=2)

Substitute n = 2 into the formula a_n = (-1)^n n^2. This gives us: a_2 = (-1)^2 * 2^2 = 4.
03

- Find the 3rd term (n=3)

Substitute n = 3 into the formula a_n = (-1)^n n^2. This gives us: a_3 = (-1)^3 * 3^2 = -9.
04

- Find the 4th term (n=4)

Substitute n = 4 into the formula a_n = (-1)^n n^2. This gives us: a_4 = (-1)^4 * 4^2 = 16.
05

- Find the 10th term (n=10)

Substitute n = 10 into the formula a_n = (-1)^n n^2. This gives us: a_{10} = (-1)^{10} * 10^2 = 100.
06

- Find the 15th term (n=15)

Substitute n = 15 into the formula a_n = (-1)^n n^2. This gives us: a_{15} = (-1)^{15} * 15^2 = -225.

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.

nth term calculation
Finding the nth term of a sequence is about determining the value of the sequence at position n. This can be done by using a sequence formula.
In the given problem, the sequence formula is \(a_n = (-1)^n n^2\).
To find any term in the sequence, we simply substitute the value of n into the formula.
  • Example: For the 10th term, we substitute n = 10 to get \(a_{10} = (-1)^{10} \times 10^2 = 100\).
  • Example: For the 15th term, we substitute n = 15 to get \(a_{15} = (-1)^{15} \times 15^2 = -225\).
By following this process, we can find any desired term in the sequence.
The formula and our understanding of exponents guide us in calculating the correct value for each term.
sequence formula
A sequence formula is a rule that defines the nth term of a sequence. It uses variables and mathematical operations to show how sequence terms are determined.
For this exercise, the given sequence formula is \(a_n = (-1)^n n^2\).
This formula tells us several important things:
  • \

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

The infinite series $$\begin{aligned} S_{\infty}=& 2+\frac{1}{2}+\frac{1}{2 \cdot 3}+\frac{1}{2 \cdot 3 \cdot 4}+\frac{1}{2 \cdot 3 \cdot 4 \cdot 5} \\ &+\frac{1}{2 \cdot 3 \cdot 4 \cdot 5 \cdot 6}+\cdots \end{aligned}$$ is not geometric, but it does have a sum. Using \(S_{1}, S_{2}, S_{3}, S_{4}, S_{5},\) and \(S_{6},\) predict the value of \(S_{\infty}\) and explain your reasoning.

During the 2015 season, Miguel Cabrera of the Detroit Tigers had a batting average of \(0.338 .\) In that season, if someone were to randomly select 5 of his "at-bats," the probability of Cabrera's getting exactly 3 hits would be the 3 rd term of the binomial expansion of \((0.338+0.662)^{5} .\) Find that term and use a calculator to estimate the probability.

Monthly Earnings. Suppose that you accepted a job for the month of February \((28\) days) under the following conditions. You will be paid \(\$ 0.01\) the first day, \(\$ 0.02\) the second, \(\$ 0.04\) the third, and so on, doubling your previous day's salary each day. How much would you earn?

The \(n\) th term of a sequence is given by \(a_{n}=n^{5}-14 n^{4}+6 n^{3}+416 n^{2}-655 n-1050 .\) Use a graphing calculator with a TABLE feature to determine which term in the sequence is 6144 .

Some sequences are given by a recursive definition. The value of the first term, \(a_{1},\) is given, and then we are told how to find any subsequent term from the term preceding it. Find the first six terms of each of the following recursively defined sequences. $$ a_{1}=1, a_{n+1}=5 a_{n}-2 $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

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.