Solved Exercises on Arithmetic Series in Algebra 2

Master arithmetic series: formulas, properties, applications, and problem-solving techniques through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic Arithmetic Series Sum
Exercise 1
Find the sum of the first 15 terms of the arithmetic sequence: 3, 7, 11, 15, ...
Definition:

Arithmetic Sequence: A sequence where each term differs from the previous by a constant called the common difference (d)

Arithmetic Series: The sum of terms in an arithmetic sequence

\(S_n = \frac{n}{2}[2a_1 + (n-1)d]\)
Sum Formula for Arithmetic Series
Solution Method:
  1. Identify the first term (a₁), common difference (d), and number of terms (n)
  2. Apply the arithmetic series formula
  3. Calculate the sum
Given
a₁=3, d=4, n=15
Formula
Sₙ=n/2[2a₁+(n-1)d]
Result
S₁₅=465
Step 1: Identify sequence parameters

a₁ = 3 (first term)

d = 7 - 3 = 4 (common difference)

n = 15 (number of terms)

Step 2: Apply the formula

S₁₅ = 15/2[2(3) + (15-1)(4)]

S₁₅ = 15/2[6 + 14(4)]

S₁₅ = 15/2[6 + 56]

S₁₅ = 15/2[62]

Step 3: Calculate the final result

S₁₅ = 15 × 31 = 465

The sum of the first 15 terms is 465
Final answer:

S₁₅ = 465

Applied rules:

• Arithmetic sequence pattern: Each term = previous term + common difference

• Sum formula: Sₙ = n/2[2a₁ + (n-1)d]

• Order of operations: Parentheses first, then multiplication and division

Tip 1: Always identify the first term and common difference first.
Tip 2: Check your arithmetic by verifying a few terms of the sequence.
2 Finding Number of Terms
Exercise 2
In an arithmetic series, the first term is 5, the common difference is 3, and the sum of the series is 375. How many terms are in the series?
Definition:

Inverse Problem: Given the sum, first term, and common difference, find the number of terms

\(S_n = \frac{n}{2}[2a_1 + (n-1)d]\)
Rearrange for n
Given
a₁=5, d=3, Sₙ=375
Quadratic
3n² + 7n - 750 = 0
Solution
n = 15
Step 1: Set up the equation

375 = n/2[2(5) + (n-1)(3)]

375 = n/2[10 + 3n - 3]

375 = n/2[7 + 3n]

Step 2: Solve for n

750 = n(7 + 3n)

750 = 7n + 3n²

3n² + 7n - 750 = 0

Step 3: Apply quadratic formula

n = [-7 ± √(49 + 9000)]/6

n = [-7 ± √9049]/6

n = [-7 ± 95.1]/6

Since n must be positive: n = 88.1/6 ≈ 15

There are 15 terms in the series
Final answer:

n = 15 terms

Applied rules:

• Algebraic manipulation: Rearrange formulas to solve for unknown variables

• Quadratic equations: Use factoring or quadratic formula when needed

• Positive solutions: Only consider positive values for count of terms

Tip 1: When solving for n, expect a quadratic equation.
Tip 2: Always verify your answer by substituting back into the original formula.
3 Alternative Sum Formula
Exercise 3
Find the sum of an arithmetic series where the first term is 8, the last term is 44, and there are 10 terms in total.
Definition:

Alternative Formula: When first term (a₁), last term (aₙ), and number of terms (n) are known

\(S_n = \frac{n}{2}(a_1 + a_n)\)
Alternative Sum Formula
Given
a₁=8, aₙ=44, n=10
Formula
Sₙ=n/2(a₁+aₙ)
Result
S₁₀=260
Step 1: Identify known values

a₁ = 8 (first term)

aₙ = 44 (last term)

n = 10 (number of terms)

Step 2: Apply the alternative formula

S₁₀ = 10/2(8 + 44)

S₁₀ = 5(52)

Step 3: Calculate the sum

S₁₀ = 260

Step 4: Verify with standard formula

First find common difference: d = (aₙ - a₁)/(n-1) = (44-8)/9 = 4

S₁₀ = 10/2[2(8) + 9(4)] = 5[16 + 36] = 5(52) = 260 ✓

The sum of the series is 260
Final answer:

S₁₀ = 260

Applied rules:

• Choice of formula: Use Sₙ = n/2(a₁ + aₙ) when first and last terms are known

• Verification: Cross-check with the standard formula for accuracy

• Relationship: aₙ = a₁ + (n-1)d connects all terms in arithmetic sequence

Tip 1: Choose the formula based on what information is given.
Tip 2: Both formulas will yield the same result when correctly applied.
Key Formulas and Properties
\(a_n = a_1 + (n-1)d\)
nth Term Formula
\(S_n = \frac{n}{2}[2a_1 + (n-1)d]\)
Standard Sum Formula
\(S_n = \frac{n}{2}(a_1 + a_n)\)
Alternative Sum Formula
Property 1
Mean Property
Each term equals average of its neighbors
Property 2
Constant Difference
aₙ - aₙ₋₁ = d (constant)
Property 3
Sum Symmetry
Terms equidistant from ends have same sum
Key Definitions:

Arithmetic Sequence: A sequence where consecutive terms differ by a constant (common difference)

Arithmetic Series: The sum of terms in an arithmetic sequence

Common Difference: The constant value added to get from one term to the next

Problem-Solving Strategy:
  1. Identify the type: Determine if it's arithmetic by checking for constant difference
  2. Gather information: List known values (a₁, d, n, aₙ, Sₙ)
  3. Choose formula: Select the most appropriate formula based on known values
  4. Solve systematically: Apply algebraic techniques to find unknowns
  5. Verify solution: Check your answer by substituting back
Common Errors: Confusing arithmetic with geometric sequences, misapplying formulas, calculation errors with negatives.
Exam Tips: Memorize both sum formulas, practice identifying arithmetic vs geometric, check for consistent differences.
Solution: Exercises 4 to 5
4 Real-world Application
Exercise 4
A theater has 25 rows of seats. The first row has 12 seats, and each subsequent row has 2 more seats than the previous row. How many total seats are in the theater?
Definition:

Real-world Application: Arithmetic series model situations with regular increases/decreases

Setup
a₁=12, d=2, n=25
Formula
Sₙ=n/2[2a₁+(n-1)d]
Result
S₂₅=950
Step 1: Identify the arithmetic sequence

Row 1: 12 seats

Row 2: 14 seats

Row 3: 16 seats

... and so on

This forms an arithmetic sequence: 12, 14, 16, ...

Step 2: Identify sequence parameters

a₁ = 12 (seats in first row)

d = 2 (common difference)

n = 25 (total number of rows)

Step 3: Apply the sum formula

S₂₅ = 25/2[2(12) + (25-1)(2)]

S₂₅ = 25/2[24 + 24(2)]

S₂₅ = 25/2[24 + 48]

S₂₅ = 25/2[72]

Step 4: Calculate the total

S₂₅ = 25 × 36 = 900

The theater has 900 total seats
Final answer:

Total seats = 900

Applied rules:

• Modeling: Translate word problems into mathematical sequences

• Pattern recognition: Identify arithmetic progression in real contexts

• Application: Use appropriate formulas to solve practical problems

Tip 1: Draw a diagram to visualize the seating pattern.
Tip 2: Verify by calculating a few terms manually.
5 Complex Series Problem
Exercise 5
The sum of the first 8 terms of an arithmetic series is 100, and the sum of the first 12 terms is 210. Find the first term and the common difference.
Definition:

System of Equations: Using multiple conditions to solve for unknown parameters

Given
S₈=100, S₁₂=210
System
8a₁+28d=100
12a₁+66d=210
Solution
a₁=5, d=2.5
Step 1: Set up equations using sum formula

For S₈ = 100: 8/2[2a₁ + (8-1)d] = 100

→ 4[2a₁ + 7d] = 100

→ 8a₁ + 28d = 100 ... (equation 1)

For S₁₂ = 210: 12/2[2a₁ + (12-1)d] = 210

→ 6[2a₁ + 11d] = 210

→ 12a₁ + 66d = 210 ... (equation 2)

Step 2: Solve the system of equations

From equation 1: 8a₁ + 28d = 100

From equation 2: 12a₁ + 66d = 210

Multiply equation 1 by 3: 24a₁ + 84d = 300

Multiply equation 2 by 2: 24a₁ + 132d = 420

Subtract: (24a₁ + 132d) - (24a₁ + 84d) = 420 - 300

48d = 120

d = 2.5

Step 3: Find the first term

Substitute d = 2.5 into equation 1:

8a₁ + 28(2.5) = 100

8a₁ + 70 = 100

8a₁ = 30

a₁ = 3.75

Step 4: Verify the solution

S₈ = 8/2[2(3.75) + 7(2.5)] = 4[7.5 + 17.5] = 4(25) = 100 ✓

S₁₂ = 12/2[2(3.75) + 11(2.5)] = 6[7.5 + 27.5] = 6(35) = 210 ✓

a₁ = 3.75, d = 2.5
Final answer:

First term: a₁ = 3.75, Common difference: d = 2.5

Applied rules:

• System solving: Use multiple conditions to form system of equations

• Elimination method: Multiply equations to eliminate variables

• Verification: Always check solutions in original conditions

Tip 1: Set up equations systematically from given information.
Tip 2: Check your work by substituting back into original conditions.
Comprehensive Guide: Arithmetic Series
\(S_n = \frac{n}{2}[2a_1 + (n-1)d]\)
Main Formula
Key definitions:

Arithmetic Sequence: A sequence where each term after the first is obtained by adding a constant (the common difference) to the preceding term

Arithmetic Series: The sum of the terms of an arithmetic sequence

Common Difference: The constant value added to get from one term to the next

Complete methodology:
  1. Identify the sequence: Verify it's arithmetic by checking for constant difference
  2. List known values: a₁ (first term), d (common difference), n (number of terms), aₙ (last term), Sₙ (sum)
  3. Select appropriate formula: Based on known and unknown values
  4. Solve systematically: Apply algebraic techniques
  5. Verify results: Check by substitution or manual calculation
Tip 1: Always verify that a sequence is arithmetic by checking the common difference.
Tip 2: The sum formula Sₙ = n/2(a₁ + aₙ) is easier when you know first and last terms.
Tip 3: Use Sₙ = n/2[2a₁ + (n-1)d] when you know the common difference.
Tip 4: Remember that arithmetic sequences model linear growth patterns.
Common errors: Forgetting to divide by 2 in sum formulas, confusing arithmetic with geometric sequences, calculation errors with negative differences.
Exam preparation: Memorize both sum formulas, practice finding nth term, work on word problems involving arithmetic sequences.
Essential formulas to know:

• nth term: aₙ = a₁ + (n-1)d

• Sum formula: Sₙ = n/2[2a₁ + (n-1)d]

• Alternative sum: Sₙ = n/2(a₁ + aₙ)

• Relationship: aₙ = a₁ + (n-1)d

Visual Understanding: Arithmetic Series
Exercise 6: Visualizing Arithmetic Series
Consider the arithmetic series: 2, 5, 8, 11, 14, 17, 20
This represents the cumulative sums of an arithmetic sequence.

Analysis: The visualization shows how arithmetic series grow quadratically with respect to the number of terms.

  • Sequence: 2, 5, 8, 11, 14, 17, 20 (arithmetic sequence)
  • Cumulative sums: 2, 7, 15, 26, 40, 57, 77 (partial sums of series)
  • Formula: Sₙ = n/2[2(2) + (n-1)(3)] = n/2(4 + 3n - 3) = n(3n + 1)/2

Questions & Answers

Question: I'm confused about when to use Sₙ = n/2[2a₁ + (n-1)d] versus Sₙ = n/2(a₁ + aₙ). Can you explain the difference?

Answer: Great question! The choice depends on what information is given:

  • Use Sₙ = n/2[2a₁ + (n-1)d] when you know the first term (a₁), the common difference (d), and the number of terms (n).
  • Use Sₙ = n/2(a₁ + aₙ) when you know the first term (a₁), the last term (aₙ), and the number of terms (n).

Both formulas give the same result, but one might be more convenient depending on the given information. If you have the common difference but not the last term, use the first formula. If you have the first and last terms, use the second formula.

Example: For sequence 3, 7, 11, 15 (4 terms):
Using first formula: S₄ = 4/2[2(3) + (4-1)(4)] = 2[6 + 12] = 36
Using second formula: S₄ = 4/2(3 + 15) = 2(18) = 36

Question: How do I know if a sequence is arithmetic? Sometimes it's not obvious from just a few terms.

Answer: To determine if a sequence is arithmetic, calculate the difference between consecutive terms:

  • Find d₁ = a₂ - a₁
  • Find d₂ = a₃ - a₂
  • Find d₃ = a₄ - a₃
  • If d₁ = d₂ = d₃ = ... then it's arithmetic with common difference d

For example, in sequence 5, 9, 13, 17:
9 - 5 = 4
13 - 9 = 4
17 - 13 = 4
Since all differences equal 4, this is an arithmetic sequence.

If the differences are not constant, it's not arithmetic. Look for patterns in the differences to identify other sequence types.

Question: What are some real-life applications of arithmetic series? Why is it important to learn this?

Answer: Arithmetic series have numerous practical applications:

  • Finance: Calculating total deposits over regular intervals with fixed increases
  • Construction: Counting materials in structures with regular patterns (like stadium seats)
  • Manufacturing: Production quantities with consistent daily increases
  • Physics: Calculating distances traveled under uniform acceleration
  • Computer Science: Algorithm complexity analysis for loops with arithmetic progressions

Learning arithmetic series develops critical thinking skills for recognizing patterns and modeling linear growth situations. It also provides foundation for more advanced mathematical concepts.

The ability to model real-world scenarios mathematically is essential for STEM careers and everyday financial planning.

Question: I often make calculation errors when working with arithmetic series. Any tips to avoid mistakes?

Answer: Here are strategies to minimize calculation errors:

  1. Write out all steps: Don't do mental calculations for complex expressions
  2. Check your arithmetic: Verify basic operations like multiplication and addition
  3. Use the distributive property carefully: Pay attention to signs when distributing
  4. Verify your answer: Calculate a few terms manually to check if your sum makes sense
  5. Work systematically: Organize your work clearly with plenty of space

For example, when calculating Sₙ = n/2[2a₁ + (n-1)d], break it down:
1. Calculate (n-1)d first
2. Add 2a₁ to that result
3. Multiply by n/2

Always double-check your final answer by substituting back into the context of the problem.

Question: How does arithmetic series relate to arithmetic sequences? Are they the same thing?

Answer: They are related but distinct concepts:

  • Arithmetic Sequence: An ordered list of numbers where each term differs from the previous by a constant (e.g., 2, 5, 8, 11, 14)
  • Arithmetic Series: The SUM of terms in an arithmetic sequence (e.g., 2 + 5 + 8 + 11 + 14 = 40)

Think of it this way: the sequence is the individual terms, while the series is their sum. The sequence follows the pattern aₙ = a₁ + (n-1)d, while the series calculates the total using sum formulas.

Every arithmetic series is based on an underlying arithmetic sequence, but the series focuses on the cumulative total rather than individual terms.

Understanding both concepts is crucial because problems often involve transitioning between the sequence (finding specific terms) and the series (finding sums).