9. Sequences, Induction, and Probability

Use the Formula for the General Term of an Arithmetic Sequence

Arithmetic Sequences

Arithmetic Sequences: A Formula for the ' n - th ' Term

In Exercises 15–22, find the indicated term of the arithmetic sequence with first term, and common difference, d. Find a60 when a₁ = 35, d = -3.

Arithmetic Progressions - Problem Solving

Find the sum of the first 22 terms of the arithmetic sequence: 5, 12, 19, 26, ...

Arithmetic Progressions - Algebra

Use the formula a_n=4+(n-1)(-7) to find the eighth term of the sequence 4, −3, −10,…

Find 1+2+3+4+...+ 100, the sum of the first 100 natural numbers.

For Exercises 45–50, 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. <img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/9f7b44cceeb4415e-1678243598836.jpg" style="width: 158px;">

In Exercises 12–15, write the first six terms of each arithmetic sequence. a1 = 3/2, d = -1/2

Find the sum of the first 50 terms of the arithmetic sequence: -10, -6, -2, 2,....

In Exercises 23–34, write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for to find the 20th term of the sequence. an= an-1 -10, a₁ = 30

Sequences, Factorials, and Summation Notation

Find the sum of the first 20 terms of the arithmetic sequence: 4, 10, 16, 22,……….

In Exercises 1–14, write the first six terms of each arithmetic sequence. a1= 5/2, d = -1/2

Finding the Sum of a Finite Arithmetic Series

In Exercises 15–22, find the indicated term of the arithmetic sequence with first term, and common difference, d. Find a50 when a₁ = 7, d = 5.

In Exercises 23–34, write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for to find the 20th term of the sequence. 1, 5, 9, 13,...

In Exercises 23–34, write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for to find the 20th term of the sequence. a1 = 9, d=2

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

In Exercises 1–14, write the first six terms of each arithmetic sequence. a1= -8, d=5

In Exercises 16–18, find the indicated term of the arithmetic sequence with first term, , and common difference, d. Find a12 when a1 = -8, d = -2

Represent an arithmetic sequence as an explicit rule

In Exercises 23–34, write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for to find the 20th term of the sequence. an = an-1 +3, a₁ = 4

In Exercises 23–34, write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for to find the 20th term of the sequence. a1=-20, d = -4

Show that the sum of the first n positive odd integers,1 +3 +5 + ··· + (2n − 1), ... is n².

In Exercises 23–34, write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for to find the 20th term of the sequence. 7,3,-1,-5,...

In Exercises 15–22, find the indicated term of the arithmetic sequence with first term, and common difference, d. Find a200 when a₁ = −40, d = 5.

Quick Intro to Arithmetic Sequences

In Exercises 1–14, write the first six terms of each arithmetic sequence. a1 = 300, d= -90

Find the sum of the first 60 positive even integers.

For Exercises 45–50, 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. <img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/f0d7c9ad47fb1c87-1678243618224.jpg" style="width: 154px;">

For Exercises 45–50, 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. <img class="fr-dib" src="https://lightcat-files.s3.amazonaws.com/problem_images/57d5d619d6dd4b91-1678243632775.jpg" style="width: 153px;">

In the sequence 21,700, 23,172, 24,644, 26,116,... which term is 314,628?

In Exercises 15–22, find the indicated term of the arithmetic sequence with first term, and common difference, d. Find a6 when a₁ = 13, d = 4.

In Exercises 1–14, write the first six terms of each arithmetic sequence. an = an-1 -10, a₁ = 30

In Exercises 19–21, write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for an to find a20, the 20th term of the sequence. -7, -3, 1, 5 ...

In Exercises 25–27, use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.  The  is a summation, refer to the textbook.

In Exercises 1–14, write the first six terms of each arithmetic sequence. an = an-1 -0.4, a₁ = 1.6

In Exercises 1–14, write the first six terms of each arithmetic sequence. a₁ = 200, d = 20

Find 3 + 6 + 9 + ... + 300, the sum of the first 100 positive multiples of 3.

Arithmetic Sequences: Finding a General Formula Given Two Terms

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

In Exercises 1–14, write the first six terms of each arithmetic sequence. an = an-1 +6, a₁ = −9

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Find the Common Difference for an Arithmetic Sequence

Write Terms of an Arithmetic Sequence

Use the Formula for the Sum of the First n Terms of an Arithmetic Sequence

Compute Empirical Probability

Compute Theoretical Probability

Find the Probability that an Event will Not Occur

Find the Probability of One Event or a Second Event Occurring

Find the Probability of One Event and a Second Event Occurring

Probability

Evaluate a Binomial Coefficient

Expand a Binomial Raised to a Power

The Binomial Theorem

Use the Fundamental Counting Principle

Distinguish Between Permutation and Combination Problems

Use the Permutations Formula

Counting Principles, Permutations, and Combinations

Find Particular Terms of a Sequence from the General Term

Use Recursion Formulas

Use Factorial Notation

Use Summation Notation

Sequences and Summation Notation

Understand the Principle of Mathematical Induction

Prove Statements Using Mathematical Induction

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Find the Common Ratio of a Geometric Sequence

Write Terms of a Geometric Sequence

Use the Formula for the General Term of a Geometric Sequence

Use the Formula for the Sum of the First n Terms of a Geometric Sequence

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