3-4 Additional Practice Arithmetic Sequences Answer Key

3-4 Additional Practice Arithmetic Sequences: Answer Key & Explained Solutions

This comprehensive guide provides detailed solutions and explanations for 3-4 additional practice problems on arithmetic sequences. Understanding arithmetic sequences is crucial for various mathematical applications, from simple calculations to complex financial modeling. This guide aims to solidify your understanding and boost your problem-solving skills. We'll cover different types of problems, emphasizing the core concepts and techniques needed to solve them efficiently and accurately.

Problem 1: Finding the nth Term

Problem: An arithmetic sequence has a first term of 5 and a common difference of 3. Find the 15th term of this sequence.

Solution:

The formula for the nth term of an arithmetic sequence is given by:

a_n = a₁ + (n-1)d

Where:

• a_n is the nth term
• a₁ is the first term
• n is the term number
• d is the common difference

In this problem:

• a₁ = 5
• d = 3
• n = 15

Substituting these values into the formula:

a₁₅ = 5 + (15-1)3 = 5 + 14 * 3 = 5 + 42 = 47

Therefore, the 15th term of the arithmetic sequence is 47.

Problem 2: Finding the Common Difference

Problem: The 5th term of an arithmetic sequence is 23 and the 10th term is 48. Find the common difference and the first term.

Solution:

We can use the formula for the nth term to set up a system of two equations:

• a₅ = a₁ + (5-1)d = 23
• a₁₀ = a₁ + (10-1)d = 48

This simplifies to:

• a₁ + 4d = 23 (Equation 1)
• a₁ + 9d = 48 (Equation 2)

Subtracting Equation 1 from Equation 2:

(a₁ + 9d) - (a₁ + 4d) = 48 - 23

5d = 25

d = 5

The common difference is 5.

Now, substitute d = 5 into Equation 1:

a₁ + 4(5) = 23

a₁ + 20 = 23

a₁ = 3

The first term is 3.

Problem 3: Sum of an Arithmetic Series

Problem: Find the sum of the first 20 terms of the arithmetic sequence: 2, 7, 12, 17...

Solution:

The formula for the sum of an arithmetic series (S_n) is:

S_n = n/2 [2a₁ + (n-1)d]

Where:

• S_n is the sum of the first n terms
• n is the number of terms
• a₁ is the first term
• d is the common difference

In this problem:

• n = 20
• a₁ = 2
• d = 7 - 2 = 5

Substituting these values:

S₂₀ = 20/2 [2(2) + (20-1)5] = 10 [4 + 19 * 5] = 10 [4 + 95] = 10 * 99 = 990

The sum of the first 20 terms is 990.

Problem 4: Finding Missing Terms

Problem: An arithmetic sequence has the terms 11, __, __, 29. Find the missing terms.

Solution:

Let the missing terms be x and y. The sequence is 11, x, y, 29. This is an arithmetic sequence, so the common difference (d) remains constant between consecutive terms.

We can express the relationship between the terms as:

• x - 11 = d
• y - x = d
• 29 - y = d

Since all these expressions are equal to d, we can create a system of equations. However, a simpler approach is to observe that there are 3 intervals between the first and last term (11 and 29). Thus, the total difference is 29 - 11 = 18. To find the common difference, we divide the total difference by the number of intervals:

d = 18 / 3 = 6

Now we can find the missing terms:

• x = 11 + d = 11 + 6 = 17
• y = x + d = 17 + 6 = 23

Therefore, the missing terms are 17 and 23. The complete sequence is 11, 17, 23, 29.

Advanced Concepts and Applications

The problems above demonstrate fundamental concepts. However, arithmetic sequences have broader applications:

• Financial Mathematics: Calculating compound interest, annuities, and loan repayments often involve arithmetic sequences or their close relative, geometric sequences.

• Physics: Analyzing uniformly accelerated motion, where the change in velocity is constant over time, utilizes arithmetic sequences.

• Computer Science: Certain algorithms and data structures rely on the properties of arithmetic sequences for efficiency.

Tips and Tricks for Solving Arithmetic Sequence Problems

• Understand the Formulas: Memorize the formulas for the nth term and the sum of an arithmetic series. Understanding the derivation of these formulas will aid in deeper comprehension.

• Identify the Key Information: Carefully extract the values of a₁, d, and n from the problem statement.

• Check Your Work: After solving a problem, verify your answer by substituting the values back into the formulas or checking the consistency within the sequence.

• Practice Regularly: Solving various problems will build your skills and confidence. Start with simpler problems and gradually progress to more complex scenarios.

• Visual Representation: Sometimes, visualizing the sequence as a list or a graph can enhance understanding and help in identifying patterns.

Further Practice Problems

To solidify your understanding, try these additional problems:

1. An arithmetic sequence has a first term of -3 and a common difference of 4. Find the 25th term.
2. The 8th term of an arithmetic sequence is 34 and the 12th term is 54. Find the common difference and the first term.
3. Find the sum of the first 30 terms of the arithmetic sequence: 1, 6, 11, 16...
4. An arithmetic sequence has terms 5, __, __, 26. Find the missing terms.
5. A student saves $5 the first week, $8 the second week, and continues to increase their savings by $3 each week. How much money will they have saved after 10 weeks?

By diligently practicing and applying the techniques outlined in this guide, you will develop proficiency in solving problems related to arithmetic sequences. Remember that consistent practice is key to mastery. Good luck!