4 7 Practice Arithmetic Sequences Form K Answers

4, 7 Practice: Mastering Arithmetic Sequences and Finding the Answers (Form K)

This comprehensive guide delves into the intricacies of arithmetic sequences, focusing on the common practice problems found in Form K materials. We'll explore the fundamental concepts, delve into practical examples, and equip you with the strategies needed to confidently solve problems related to arithmetic sequences. This guide is designed to be thorough, providing a deep understanding rather than just quick solutions. We will cover various problem types and show different approaches to find the answers.

Understanding Arithmetic Sequences: The Foundation

An arithmetic sequence is a series of numbers where the difference between consecutive terms remains constant. This constant difference is called the common difference, often denoted as 'd'. Understanding this core principle is crucial for solving any problem involving arithmetic sequences.

Key Elements:

• First Term (a₁): The initial number in the sequence.
• Common Difference (d): The constant difference between consecutive terms.
• nth Term (aₙ): The term at the nth position in the sequence.

Formula for the nth Term:

The most important formula for working with arithmetic sequences is the formula for finding the nth term:

aₙ = a₁ + (n-1)d

This formula allows you to calculate any term in the sequence, given the first term, common difference, and the position of the term.

Example 1: Finding the nth Term

Let's say we have an arithmetic sequence with a first term (a₁) of 4 and a common difference (d) of 3. What is the 10th term (a₁₀)?

Using the formula:

a₁₀ = 4 + (10-1)3 = 4 + 27 = 31

Therefore, the 10th term in this sequence is 31.

Form K Practice Problems: A Deep Dive

Form K often presents a variety of problems testing your understanding of arithmetic sequences. Let's tackle some common types:

Problem Type 1: Finding the Common Difference

These problems will typically provide several consecutive terms of an arithmetic sequence and ask you to find the common difference.

Example 2:

Find the common difference in the arithmetic sequence: 7, 11, 15, 19...

Solution:

Simply subtract any consecutive term from the following term:

11 - 7 = 4 15 - 11 = 4 19 - 15 = 4

The common difference (d) is 4.

Problem Type 2: Finding a Specific Term

This is where the formula aₙ = a₁ + (n-1)d becomes invaluable. These problems will give you the first term, common difference, and the position (n) and ask you to find the term at that position.

Example 3:

An arithmetic sequence has a first term of 2 and a common difference of 5. Find the 8th term.

Solution:

Using the formula:

a₈ = 2 + (8-1)5 = 2 + 35 = 37

The 8th term is 37.

Problem Type 3: Finding the First Term

Sometimes, you might be given a term other than the first and the common difference, and asked to find the first term.

Example 4:

The 5th term of an arithmetic sequence is 22, and the common difference is 4. Find the first term.

Solution:

We can rearrange the formula: aₙ = a₁ + (n-1)d to solve for a₁:

a₁ = aₙ - (n-1)d

Substituting the given values:

a₁ = 22 - (5-1)4 = 22 - 16 = 6

The first term is 6.

Problem Type 4: Finding the Number of Terms

These problems require you to find 'n', the number of terms in a given sequence. You'll be given the first term, common difference, and the last term.

Example 5:

An arithmetic sequence has a first term of 3, a common difference of 2, and a last term of 27. How many terms are in the sequence?

Solution:

We use the formula aₙ = a₁ + (n-1)d and solve for n:

27 = 3 + (n-1)2 24 = (n-1)2 12 = n-1 n = 13

There are 13 terms in the sequence.

Problem Type 5: Finding the Sum of an Arithmetic Sequence

The sum of an arithmetic sequence can be found using the following formula:

Sₙ = n/2 [2a₁ + (n-1)d] or Sₙ = n/2 (a₁ + aₙ)

where Sₙ is the sum of the first n terms.

Example 6:

Find the sum of the first 10 terms of the arithmetic sequence with a first term of 1 and a common difference of 3.

Solution:

Using the first formula:

S₁₀ = 10/2 [2(1) + (10-1)3] = 5 [2 + 27] = 5(29) = 145

Using the second formula (we need to find a₁₀ first):

a₁₀ = 1 + (10-1)3 = 28 S₁₀ = 10/2 (1 + 28) = 5(29) = 145

The sum of the first 10 terms is 145.

Advanced Problem Solving Techniques

Form K might include more complex problems requiring a combination of the techniques discussed above. Let's explore a few scenarios:

Scenario 1: Sequences within Sequences

Imagine a problem where you have an arithmetic sequence, and then another sequence formed by taking the sum of consecutive terms of the original sequence.

Example 7:

Given the arithmetic sequence 2, 5, 8, 11, 14..., find the 5th term of the sequence formed by summing consecutive terms of the original sequence (e.g., 2+5, 5+8, 8+11...).

Solution:

First, find the sums: 7, 13, 19, 25... This is a new arithmetic sequence. The common difference is 6. The first term is 7. We want the 5th term.

Using aₙ = a₁ + (n-1)d :

a₅ = 7 + (5-1)6 = 7 + 24 = 31

The 5th term of the sum sequence is 31.

Scenario 2: Word Problems Involving Arithmetic Sequences

Real-world applications often involve arithmetic sequences.

Example 8:

A person saves $50 in the first week, $55 in the second week, $60 in the third week, and so on. How much will they have saved after 12 weeks?

Solution:

This is an arithmetic sequence with a₁ = 50 and d = 5. We want to find the sum of the first 12 terms (S₁₂).

First, find a₁₂: a₁₂ = 50 + (12-1)5 = 105

Then, use the sum formula: S₁₂ = 12/2 (50 + 105) = 6(155) = 930

They will have saved $930 after 12 weeks.

Mastering Arithmetic Sequences: Key Takeaways

By understanding the core concepts of arithmetic sequences – the first term, common difference, and the formula for the nth term – you can confidently tackle a wide range of problems. Remember to practice regularly, working through various problem types and scenarios. The more you practice, the more comfortable you'll become in identifying the appropriate formulas and techniques to apply. Consistent practice and a clear understanding of the fundamental principles are the keys to success in mastering arithmetic sequences and achieving high scores on Form K. Don't be afraid to break down complex problems into smaller, manageable steps. Remember to always double-check your work to minimize errors. With focused effort and persistent practice, you will master arithmetic sequences.