IB Mathematics AI SL Geometric sequences and series Study Notes - New Syllabus

IB Mathematics AI SL Geometric sequences and series Study Notes

LEARNING OBJECTIVE

Use of the formulae for the nth term and the sum of the first n terms of the sequence. Use of sigma notation for sums of geometric sequences.

Key Concepts:

geometric sequences and series.

GEOMETRIC SEQUENCE (G.S.)

♦ Definition
A sequence where the ratio ($ r $) between consecutive terms is constant.
Example: $ u_1 = 5 $, $ r = 2 $ $→ 5, 10, 20, 40, …$

♦ General Formula:
$ u_n = u_1 r^{n-1} $

♦ Consecutive Terms:
For $ a, x, b $ in G.S., $ x^2 = ab $.

♦ Sum of First $ n $ Terms ($ S_n $):
$ S_n = \frac{u_1(r^n – 1)}{r – 1} \quad (r \neq 1) $

♦ Sum to infinity (when $ |r| < 1 $):

$S_\infty = \frac{a}{1 – r}$

Example

(i) $ u_1 = 1 $, $ r = -2 $

(ii) $ u_1 = 1 $, $ r = \frac{1}{2} $

▶️Answer/Explanation

Solution:

(i) $→ 1, -2, 4, -8, …$

(ii) $→ 1,$ $ \frac{1}{2} $, $ \frac{1}{4} $, …

Example

$ u_1 = 3 $, $ r = 2 $

▶️Answer/Explanation

Solution:

$→$ $ u_{100} = 3 \times 2^{99} $

Example

Sum of $ 2 + 4 + 8 + \cdots + 2^{10} $

▶️Answer/Explanation

Solution:

$ S_{10} = 2046 $

Example

For 10, $ x $, 90 in G.S.,

▶️Answer/Explanation

Solution:

$ x = \pm 30 $.

APPLICATIONS

♦Real-Life Applications of Geometric Sequences

Scenario	Description
Population growth	Populations growing by a fixed percentage each year follow geometric growth.
Spread of disease	Infection rates that double or triple over time show geometric increase.
Salary changes	Annual raises or deductions by a fixed percentage are modeled geometrically.
Depreciation	Value of a car or asset decreasing by a fixed % yearly.

Disease Spread

A virus spreads to twice as many people each day. On day 1, 5 people are infected.

$ a = 5 $, $ r = 2 $

Day 4 infected:?

Total infected after 5 days:?

▶️Answer/Explanation

Solution:

Day 4 infected:
$u_4 = 5 \cdot 2^{3} = 5 \cdot 8 = \rm{40}$

Total infected after 5 days:
$S_5 = 5 \cdot \frac{1 – 2^5}{1 – 2} = 5 \cdot \frac{1 – 32}{-1} = 5 \cdot 31 = \rm{155}$

Salary Increase

A worker earns \$30,000 and receives a 5% raise annually.

$ a = 30{,}000 $, $ r = 1.05 $

After 3 years:?

Total earned over 5 years:?

▶️Answer/Explanation

Solution:

After 3 years:

$u_3 = 30{,}000 \cdot (1.05)^2 = 30{,}000 \cdot 1.1025 = \rm{\$33,075}$

Total earned over 5 years:
$S_5 = 30{,}000 \cdot \frac{1 – (1.05)^5}{1 – 1.05} \approx \rm{\$165,765}$

Depreciation of Car

A car worth \$20,000 depreciates by 20% per year.

$ a = 20{,}000 $, $ r = 0.8 $
Value after 4 years:?

▶️Answer/Explanation

Solution:

Value after 4 years:
$u_5 = 20{,}000 \cdot (0.8)^4 = 20{,}000 \cdot 0.4096 = \rm{\$8,192}$

Interpretation Notes

If $ r > 1 $ → exponential growth
If $ 0 < r < 1 $ → exponential decay

IB Mathematics AI SL Geometric sequences and series Study Notes - New Syllabus

IB Mathematics AI SL Geometric sequences and series Study Notes

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