Ch. 8 - Sequences, Induction, and Probability

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

Problem 44

Chapter 9, Problem 44

Find the sum of the odd integers between 30 and 54.

Verified step by step guidance

Identify the range of odd integers between 30 and 54. Since 30 is even, the first odd integer greater than 30 is 31, and since 54 is even, the last odd integer less than 54 is 53.

List the odd integers between 30 and 54: 31, 33, 35, ..., 53. These form an arithmetic sequence where the first term

a = 31

and the last term

l = 53

Determine the common difference

d

of the sequence, which is 2 because odd numbers increase by 2.

Calculate the number of terms

n

in the sequence using the formula for the nth term of an arithmetic sequence:

l = a + (n - 1) d

. Solve for

n

Use the formula for the sum of an arithmetic sequence:

S_n = \(\frac{n}{2}\) (a + l)

. Substitute the values of

n

a

, and

l

to express the sum.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Identifying Odd Integers

Odd integers are whole numbers that are not divisible by 2, typically ending with digits 1, 3, 5, 7, or 9. To find odd integers between two numbers, list all numbers in that range and select those that are odd.

Arithmetic Sequences

An arithmetic sequence is a list of numbers with a constant difference between consecutive terms. Odd integers form an arithmetic sequence with a common difference of 2, which helps in efficiently finding sums without listing all terms.

Sum of an Arithmetic Sequence

The sum of an arithmetic sequence can be found using the formula S = n/2 * (first term + last term), where n is the number of terms. This formula simplifies adding many numbers by using only the first and last terms and the count of terms.

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Textbook Question

Write out the first three terms and the last term. Then use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.

$\sum_{i = 1}^{17} (5 i + 3)$

Textbook Question

Find the term indicated in each expansion. (x²+ y³)⁸; sixth term

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Textbook Question

Find the sum of the even integers between 21 and 45.

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Textbook Question

In Exercises 45-46, it is equally probable that the pointer on the spinner shown will land on any one of the eight regions, numbered 1 through 8. If the pointer lands on a borderline, spin again. Find the probability that the pointer will stop on an odd number or a number less than 6.

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Textbook Question

Use the formula for the sum of the first n terms of a geometric sequence to find the indicated sum.

$\sum_{i = 1}^{6} 5^{i}$

Textbook Question

Express each sum using summation notation. Use $1$ as the lower limit of summation and $i$ for the index of summation. $2 + 2^{2} + 2^{3} + \dots + 2^{11}$