Geometric Series
Master finite and infinite geometric series, understand convergence conditions, and apply these concepts to real-world problems in HSC Advanced Mathematics.
Finite Geometric Series
A geometric series is the sum of terms in a geometric sequence, where each term is obtained by multiplying the previous term by a constant common ratio r. The sum of the first n terms is:
Sn = a(rn − 1)r − 1, r ≠ 1
Where a is the first term, r is the common ratio, and n is the number of terms.
An equivalent form, useful when |r| < 1, is:
Sn = a(1 − rn)1 − r
Example: Find the sum of 2 + 6 + 18 + 54 + 162.
Here a = 2, r = 3, n = 5.
S5 = 2(35 − 1)3 − 1 = 2(243 − 1)/2 = 242
Infinite Geometric Series
When the common ratio satisfies |r| < 1, the geometric series converges to a finite sum as the number of terms approaches infinity. This is called the limiting sum or sum to infinity:
S∞ = a/1 − r, |r| < 1
The series converges only when the absolute value of the common ratio is less than 1.
If |r| ≥ 1, the series diverges and does not have a finite sum. The terms grow larger (or oscillate without settling) and the partial sums increase without bound.
Example: Find the limiting sum of 8 + 4 + 2 + 1 + ...
Here a = 8, r = 1/2. Since |1/2| < 1, the series converges.
S∞ = 8/1 − 1/2 = 8/1/2 = 16
Applications and Convergence
Geometric series appear in compound interest, depreciation, drug dosage accumulation, and recurring decimals.
Recurring decimal: Express 0.333... as a fraction.
0.333... = 3/10 + 3/100 + 3/1000 + ... This is a geometric series with a = 3/10, r = 1/10.
S∞ = 3/10/1 − 1/10 = 3/10/9/10 = 3/9 = 1/3
Depreciation: A car worth $40,000 depreciates by 15% each year. Find its total value loss over 5 years.
Year 1 loss: 40000 × 0.15 = 6000. Each subsequent loss is 0.85 times the previous.
Total loss = 6000 × 1 − 0.8551 − 0.85 = 6000 × 1 − 0.4437/0.15 ≈ $22,252
Key Vocabulary
Common Ratio
The constant factor r by which each term is multiplied to obtain the next term in a geometric sequence.
Convergence
An infinite series converges when the partial sums approach a finite value. For geometric series, this occurs when |r| < 1.
Limiting Sum
The value that the sum of an infinite convergent geometric series approaches: S∞ = a/(1 − r).
Divergence
An infinite series diverges when the partial sums do not approach a finite value, occurring when |r| ≥ 1.
Worked Examples
Find the sum of the first 8 terms of the geometric series 3 + 6 + 12 + 24 + ...
Step 1: Identify a = 3, r = 2, n = 8.
Step 2: S8 = 3(28 − 1)2 − 1 = 3(256 − 1) = 3 × 255.
Answer: S8 = 765
Find the limiting sum of 12 − 6 + 3 − 1.5 + ...
Step 1: Identify a = 12, r = −1/2. Since |−1/2| = 0.5 < 1, the series converges.
Step 2: S∞ = 12/1 − (−1/2) = 12/3/2.
Answer: S∞ = 8
For what values of x does the series 1 + (x − 2) + (x − 2)2 + ... converge? Find S∞.
Step 1: Here a = 1 and r = x − 2. For convergence: |x − 2| < 1, so 1 < x < 3.
Step 2: S∞ = 1/1 − (x − 2) = 1/3 − x.
Answer: The series converges for 1 < x < 3, and S∞ = 1/(3 − x).
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
Find S6 for the geometric series with a = 5 and r = 2.
Question 2
What is the limiting sum of 18 + 6 + 2 + 2/3 + ...?
Question 3
Which of the following geometric series converges?
Question 4
A geometric series has a = 10 and S∞ = 40. What is r?
Question 5
Express 0.272727... as a fraction using geometric series.
Key Concepts Summary
- ● The finite geometric sum is Sn = a(rn − 1)/(r − 1) for r ≠ 1.
- ● An infinite geometric series converges when |r| < 1, with limiting sum S∞ = a/(1 − r).
- ● If |r| ≥ 1, the infinite geometric series diverges.
- ● Geometric series model compound growth, depreciation, and recurring decimals.
- ● Always verify the convergence condition before applying the limiting sum formula.