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Arithmetic and Geometric Series

Finding sums of sequences

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Series and Summation

What is a Series?

A series is the sum of the terms in a sequence.

Example: 2+4+6+8+10=302 + 4 + 6 + 8 + 10 = 30

Arithmetic Series

Sum of an arithmetic sequence: Sn=n(a1+an)2S_n = \frac{n(a_1 + a_n)}{2}

or

Sn=n[2a1+(n−1)d]2S_n = \frac{n[2a_1 + (n-1)d]}{2}

where nn = number of terms

Geometric Series

Sum of a geometric sequence: Sn=a1⋅1−rn1−rS_n = a_1 \cdot \frac{1 - r^n}{1 - r}

(when r≠1r \neq 1)

Infinite Geometric Series

If ∣r∣<1|r| < 1, the infinite series has a sum: S=a11−rS = \frac{a_1}{1 - r}

Example: 1+12+14+18+...1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...

S=11−12=112=2S = \frac{1}{1 - \frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2

Sigma Notation

∑i=1nai=a1+a2+a3+...+an\sum_{i=1}^{n} a_i = a_1 + a_2 + a_3 + ... + a_n

Read as: "the sum from i=1i = 1 to nn of aia_i"

📚 Practice Problems

1Problem 1easy

❓ Question:

Find the sum: 2+5+8+11+142 + 5 + 8 + 11 + 14

💡 Show Solution

This is an arithmetic series with:

  • a1=2a_1 = 2
  • an=14a_n = 14
  • n=5n = 5 terms

Use the formula: Sn=n(a1+an)2S_n = \frac{n(a_1 + a_n)}{2}

S5=5(2+14)2=5(16)2=802=40S_5 = \frac{5(2 + 14)}{2} = \frac{5(16)}{2} = \frac{80}{2} = 40

Answer: 4040

2Problem 2medium

❓ Question:

Find the sum of the first 6 terms: 3,6,12,24,...3, 6, 12, 24, ...

💡 Show Solution

This is a geometric series with:

  • a1=3a_1 = 3
  • r=2r = 2
  • n=6n = 6

Use the formula: Sn=a1⋅1−rn1−rS_n = a_1 \cdot \frac{1 - r^n}{1 - r}

S6=3⋅1−261−2S_6 = 3 \cdot \frac{1 - 2^6}{1 - 2} =3⋅1−64−1= 3 \cdot \frac{1 - 64}{-1} =3⋅−63−1= 3 \cdot \frac{-63}{-1} =3⋅63=189= 3 \cdot 63 = 189

Answer: 189189

3Problem 3hard

❓ Question:

Find the sum of the infinite series: 8+4+2+1+...8 + 4 + 2 + 1 + ...

💡 Show Solution

This is an infinite geometric series with:

  • a1=8a_1 = 8
  • r=48=12r = \frac{4}{8} = \frac{1}{2}

Since ∣r∣=12<1|r| = \frac{1}{2} < 1, the series converges.

Use the formula: S=a11−rS = \frac{a_1}{1 - r}

S=81−12S = \frac{8}{1 - \frac{1}{2}} =812= \frac{8}{\frac{1}{2}} =16= 16

Answer: 1616

Explain using:

⚠️ Common Mistakes: Arithmetic and Geometric Series

Avoid these 3 frequent errors

🌍 Real-World Applications: Arithmetic and Geometric Series

See how this math is used in the real world

📝 Worked Example: Solving a Quadratic by Factoring

Problem:

Solve x2−5x+6=0x^2 - 5x + 6 = 0.

2Factor the quadratic
3Set each factor equal to zero
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📌 Related Topics in Sequences and Series

Geometric SequencesGeometric Sequences and SeriesInfinite Series

❓ Frequently Asked Questions

What is Arithmetic and Geometric Series?▾
Finding sums of sequences
How can I study Arithmetic and Geometric Series effectively?▾
Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 3 problems provided, checking solutions as you go. Regular review and active practice are key to retention.
Is this Arithmetic and Geometric Series study guide free?▾
Yes — all study notes, flashcards, and practice problems for Arithmetic and Geometric Series on Study Mondo are free to access. No account is needed.
What course covers Arithmetic and Geometric Series?▾
Arithmetic and Geometric Series is part of the Algebra 2 course on Study Mondo, specifically in the Sequences and Series section. You can explore the full course for more related topics and practice resources.
Are there practice problems for Arithmetic and Geometric Series?▾
Yes, this page includes 3 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.