Solved Exercises on Prime and Composite Numbers Chart

Master prime and composite numbers: identifying primes, finding composites, and understanding factors through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Prime Number Identification
Exercise 1
Using the prime and composite numbers chart, identify if 13 is a prime number and explain why.
Definition:

Prime number: A number greater than 1 that has exactly two factors: 1 and itself

Prime identification method:
  1. Locate the number on the prime and composite numbers chart
  2. List all possible factors of the number
  3. Count the total number of factors
  4. If there are exactly 2 factors, it's prime
  5. If there are more than 2 factors, it's composite
Number
13
Factors
1, 13
Type
Prime
Step 1: Locate 13 on the chart

Looking at the prime and composite numbers chart, 13 is highlighted in yellow (prime number)

Step 2: List all factors of 13

We need to find all numbers that divide 13 evenly (with no remainder)

13 ÷ 1 = 13 (remainder 0) ✓

13 ÷ 2 = 6 remainder 1 (not a factor)

13 ÷ 3 = 4 remainder 1 (not a factor)

13 ÷ 4 = 3 remainder 1 (not a factor)

13 ÷ 5 = 2 remainder 3 (not a factor)

13 ÷ 6 = 2 remainder 1 (not a factor)

13 ÷ 7 = 1 remainder 6 (not a factor)

13 ÷ 8 = 1 remainder 5 (not a factor)

13 ÷ 9 = 1 remainder 4 (not a factor)

13 ÷ 10 = 1 remainder 3 (not a factor)

13 ÷ 11 = 1 remainder 2 (not a factor)

13 ÷ 12 = 1 remainder 1 (not a factor)

Step 3: Count the factors

Only 1 and 13 divide 13 evenly

Total factors: 2

Step 4: Determine if prime or composite

Since 13 has exactly 2 factors (1 and 13), it is a prime number

13 is PRIME
Final answer:

Yes, 13 is a prime number because it has exactly two factors: 1 and 13

Applied rules:

• Prime Definition: Exactly two factors: 1 and the number itself

• Factor Testing: Check divisibility by all numbers less than the square root

• Counting Factors: Prime numbers have exactly 2 factors

2 Composite Number Identification
Exercise 2
Using the chart, determine if 24 is a composite number and list all its factors.
Definition:

Composite number: A number greater than 1 that has more than two factors

Number
24
Factors
1,2,3,4,6,8,12,24
Type
Composite
Step 1: Locate 24 on the chart

Looking at the prime and composite numbers chart, 24 is highlighted in blue (composite number)

Step 2: Find all factors of 24

We need to find all numbers that divide 24 evenly (with no remainder)

24 ÷ 1 = 24 (remainder 0) ✓

24 ÷ 2 = 12 (remainder 0) ✓

24 ÷ 3 = 8 (remainder 0) ✓

24 ÷ 4 = 6 (remainder 0) ✓

24 ÷ 6 = 4 (remainder 0) ✓

24 ÷ 8 = 3 (remainder 0) ✓

24 ÷ 12 = 2 (remainder 0) ✓

24 ÷ 24 = 1 (remainder 0) ✓

Step 3: List all factors in order

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Step 4: Count the factors

Total factors: 8 (more than 2)

Step 5: Determine if prime or composite

Since 24 has more than 2 factors, it is a composite number

24 is COMPOSITE with factors: 1, 2, 3, 4, 6, 8, 12, 24
Final answer:

Yes, 24 is a composite number because it has more than two factors: 1, 2, 3, 4, 6, 8, 12, 24

Applied rules:

• Composite Definition: More than two factors

• Factor Pairs: When you find one factor, you also find its pair

• Systematic Approach: Check divisors up to the square root of the number

3 Finding Specific Terms
Exercise 3
Using the chart, find the 12th term of the arithmetic sequence: 8, 13, 18, 23, ...
Definition:

nth term: A formula that allows finding any term in a sequence without listing all previous terms

Given
a₁=8, d=5
Formula
aₙ = 8 + (n-1)5
12th Term
a₁₂ = 63
Step 1: Identify the first term and common difference

First term (a₁) = 8

Common difference (d) = 13 - 8 = 5

Step 2: Apply the arithmetic sequence formula

Formula: aₙ = a₁ + (n-1)d

aₙ = 8 + (n-1)5

Step 3: Substitute n = 12 to find the 12th term

a₁₂ = 8 + (12-1)5

a₁₂ = 8 + 11×5

a₁₂ = 8 + 55 = 63

Step 4: Verify the answer

Check: 8, 13, 18, 23, 28, 33, 38, 43, 48, 53, 58, 63 (each adds 5) ✓

12th term: 63
Final answer:

The 12th term of the sequence is 63

Applied rules:

• Arithmetic Formula: aₙ = a₁ + (n-1)d

• Substitution: Replace variables with known values

• Order of Operations: Calculate (n-1) first, then multiply by d

Prime and Composite Numbers Chart Summary & Methods
\(a_n = a_1 + (n-1)d\)
Arithmetic Sequence Formula
nth Term
\(a_n = a_1 + (n-1)d\)
d is common difference
Sum of n terms
\(S_n = \frac{n}{2}(a_1 + a_n)\)
or \(\frac{n}{2}[2a_1 + (n-1)d]\)
Common Difference
\(d = a_{n+1} - a_n\)
Difference between consecutive terms
Key definitions:

Arithmetic Sequence: A sequence where each term increases by a constant difference

Common Difference (d): The constant value added to each term to get the next term

First Term (a₁): The initial value in the sequence

nth Term (aₙ): The term at position n in the sequence

Consecutive Terms: Terms that follow one after another in the sequence

Finite Sequence: Has a limited number of terms

Infinite Sequence: Continues indefinitely

Arithmetic sequence solving methods:
  1. Identify the sequence type: Check if differences between terms are constant
  2. Find the common difference: Subtract consecutive terms
  3. Use the appropriate formula: For nth term, sum, or missing terms
  4. Substitute known values: Replace variables in the formula
  5. Calculate: Perform the mathematical operations
  6. Verify: Check that your answer makes sense in the context
Tip 1: Always check that the difference between consecutive terms is constant.
Tip 2: When finding missing terms, use the common difference to move forward or backward.
Tip 3: For the nth term formula, remember to subtract 1 from the position (n-1).
Tip 4: Verify your answer by checking if it fits the pattern of the sequence.
Common errors: Forgetting to subtract 1 in the formula (n-1), miscalculating the common difference, not verifying the answer, confusing arithmetic with geometric sequences.
Key notes: The common difference can be positive (increasing sequence) or negative (decreasing sequence), always verify your calculations by checking multiple consecutive terms.
Essential formulas:

• nth term: \(a_n = a_1 + (n-1)d\)

• Sum of n terms: \(S_n = \frac{n}{2}(a_1 + a_n)\)

• Common difference: \(d = a_{n+1} - a_n\)

• Alternative sum: \(S_n = \frac{n}{2}[2a_1 + (n-1)d]\)

Solution: Exercises 4 to 5
4 Decreasing Arithmetic Sequence
Exercise 4
Find the 8th term of the arithmetic sequence: 50, 43, 36, 29, ...
Definition:

Decreasing arithmetic sequence: An arithmetic sequence with a negative common difference

Given
a₁=50, d=-7
Formula
aₙ = 50 + (n-1)(-7)
8th Term
a₈ = -3
Step 1: Find the common difference

43 - 50 = -7

36 - 43 = -7

29 - 36 = -7

Common difference (d) = -7

Step 2: Identify the first term

First term (a₁) = 50

Step 3: Apply the arithmetic sequence formula

Formula: aₙ = a₁ + (n-1)d

aₙ = 50 + (n-1)(-7)

Step 4: Find the 8th term

Substitute n = 8: a₈ = 50 + (8-1)(-7)

a₈ = 50 + 7×(-7)

a₈ = 50 - 49 = 1

8th term: 1
Final answer:

The 8th term of the sequence is 1

Applied rules:

• Negative Common Difference: Results in a decreasing sequence

• Same Formula: Arithmetic formula works for both increasing and decreasing sequences

• Sign Matters: Negative differences are subtracted in calculations

5 Sum of Arithmetic Sequence
Exercise 5
Find the sum of the first 10 terms of the arithmetic sequence: 6, 10, 14, 18, ...
Definition:

Sum of arithmetic sequence: The total of all terms in the sequence up to a certain position

Given
a₁=6, d=4, n=10
10th Term
a₁₀ = 42
Sum
S₁₀ = 240
Step 1: Identify known values

First term (a₁) = 6

Common difference (d) = 10 - 6 = 4

Number of terms (n) = 10

Step 2: Find the 10th term using the formula

aₙ = a₁ + (n-1)d

a₁₀ = 6 + (10-1)4 = 6 + 36 = 42

Step 3: Apply the sum formula

Formula: Sₙ = n/2(a₁ + aₙ)

S₁₀ = 10/2(6 + 42) = 5 × 48 = 240

Step 4: Verify using alternative formula

Formula: Sₙ = n/2[2a₁ + (n-1)d]

S₁₀ = 10/2[2(6) + (10-1)4] = 5[12 + 36] = 5 × 48 = 240 ✓

Sum of first 10 terms: 240
Final answer:

The sum of the first 10 terms is 240

Applied rules:

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

• Two Options: Use either formula depending on available information

• Verification: Both formulas should yield the same result

Complete Guide to Arithmetic Sequences
\(a_n = a_1 + (n-1)d\)
Arithmetic Sequence Formula
Key definitions:

Arithmetic Sequence: A sequence where each term increases by a constant difference (d)

Common Difference (d): The fixed number added to each term to get the next term

First Term (a₁): The initial value in the sequence

nth Term (aₙ): The term at position n in the sequence

Increasing Sequence: When d > 0 (positive common difference)

Decreasing Sequence: When d < 0 (negative common difference)

Finite Sequence: Has a limited number of terms

Infinite Sequence: Continues indefinitely

Arithmetic sequence solving methodology:
  1. Identify the sequence: Check if differences between consecutive terms are constant
  2. Find the common difference: Calculate d = a₂ - a₁
  3. Determine what's needed: nth term, sum, or missing terms
  4. Select the appropriate formula: Use the correct formula for the task
  5. Substitute values: Replace variables with known quantities
  6. Calculate: Perform the mathematical operations carefully
  7. Verify: Check that your answer makes sense in the sequence context
Tip 1: Always verify that the difference between consecutive terms is truly constant.
Tip 2: Remember to subtract 1 in the formula: aₙ = a₁ + (n-1)d, not a₁ + nd.
Tip 3: For decreasing sequences, the common difference will be negative.
Tip 4: When finding the sum, you can use either formula depending on which values you have.
Tip 5: If you know two terms in the sequence, you can find the common difference.
Tip 6: Always double-check your work by verifying the pattern continues correctly.
Common errors: Forgetting to subtract 1 in the formula (n-1), miscalculating the common difference, not paying attention to positive/negative signs, confusing arithmetic with geometric sequences, making calculation errors when substituting values, not verifying the final answer.
Exam preparation: Memorize the arithmetic sequence formulas, practice identifying sequences, master finding missing terms, understand how to calculate sums, practice both increasing and decreasing sequences, work with word problems involving arithmetic sequences.
Essential formulas:

• nth term: \(a_n = a_1 + (n-1)d\)

• Sum of n terms: \(S_n = \frac{n}{2}(a_1 + a_n)\)

• Common difference: \(d = a_{n+1} - a_n\)

• Alternative sum: \(S_n = \frac{n}{2}[2a_1 + (n-1)d]\)

• Middle term (for odd n): \(\frac{a_1 + a_n}{2}\)

Arithmetic Sequences Visualization
Exercise 6: Comparing Different Sequences
Compare different arithmetic sequences:
\(f_1(n) = 3n + 2\) (increasing)
\(f_2(n) = 50 - 4n\) (decreasing)
\(f_3(n) = 10 + 0n\) (constant)

Analysis: The chart shows how different arithmetic sequences behave.

  • \(f_1(n) = 3n + 2\): Increasing sequence (positive slope)
  • \(f_2(n) = 50 - 4n\): Decreasing sequence (negative slope)
  • \(f_3(n) = 10\): Constant sequence (zero slope)

Questions & Answers

Question: How do I know if a sequence is arithmetic or not?

Answer: To determine if a sequence is arithmetic, follow these steps:

  1. Subtract the first term from the second term
  2. Subtract the second term from the third term
  3. Continue this process for all consecutive terms
  4. If all the differences are the same number, the sequence is arithmetic

Example: 5, 9, 13, 17, 21

  • 9 - 5 = 4
  • 13 - 9 = 4
  • 17 - 13 = 4
  • 21 - 17 = 4

Since all differences equal 4, this is an arithmetic sequence with common difference d = 4.

Question: What does the formula aₙ = a₁ + (n-1)d mean?

Answer: This is the arithmetic sequence formula that helps find any term in the sequence:

  • aₙ: The term you want to find (at position n)
  • a₁: The first term in the sequence
  • n: The position of the term you want to find
  • d: The common difference between terms
  • (n-1): How many times you need to add the common difference

Example: For sequence 2, 5, 8, 11... (a₁=2, d=3)

  • To find the 4th term: a₄ = 2 + (4-1)×3 = 2 + 9 = 11
  • This means: start at first term (2), add the difference (3) three times (4-1=3)

This formula is much faster than listing all terms up to the one you want!

Question: Can arithmetic sequences decrease instead of increase?

Answer: Yes! Arithmetic sequences can be decreasing when the common difference is negative:

  • Increasing sequence: Common difference is positive (d > 0)
  • Decreasing sequence: Common difference is negative (d < 0)
  • Constant sequence: Common difference is zero (d = 0)

Examples:

  • Increasing: 3, 7, 11, 15... (d = +4)
  • Decreasing: 20, 15, 10, 5... (d = -5)
  • Constant: 7, 7, 7, 7... (d = 0)

The same formulas work for all types of arithmetic sequences, just pay attention to the sign of the common difference!

Question: How do I find the sum of an arithmetic sequence?

Answer: There are two formulas for finding the sum of the first n terms of an arithmetic sequence:

Formula 1: Sₙ = n/2(a₁ + aₙ)

  • Use this when you know the first term (a₁), last term (aₙ), and number of terms (n)

Formula 2: Sₙ = n/2[2a₁ + (n-1)d]

  • Use this when you know the first term (a₁), common difference (d), and number of terms (n)

Example: Find sum of first 5 terms of sequence 2, 5, 8, 11, 14

  • Using Formula 1: S₅ = 5/2(2 + 14) = 5/2(16) = 40
  • Using Formula 2: S₅ = 5/2[2(2) + (5-1)3] = 5/2[4 + 12] = 5/2(16) = 40

Both formulas give the same result - choose the one that uses values you already know!

Question: What's the best way to practice arithmetic sequences for a quizlet?

Answer: Here's an effective practice strategy for arithmetic sequences:

  1. Memorize formulas: Know the nth term formula and sum formulas by heart
  2. Practice identification: Work on recognizing arithmetic sequences among other patterns
  3. Master calculations: Practice finding terms, common differences, and sums
  4. Work with different scenarios: Practice both increasing and decreasing sequences
  5. Create flashcards: Make cards with formulas, examples, and common mistakes to avoid
  6. Take practice tests: Simulate quiz conditions to build confidence

Focus on understanding the concepts, not just memorizing formulas. Practice with various types of problems to be prepared for anything on the quizlet!

Remember: Consistent practice and understanding the underlying concepts are key to success!