Sequences (applications)

Sequences are ordered lists of numbers following a specific pattern or rule. In Mathematics: Applications & Interpretation, we focus on arithmetic sequences (constant difference) and geometric sequences (constant ratio), with emphasis on real-world applications.

Arithmetic Sequences: Each term differs from the previous by a constant value called the common difference (d).

• General term formula: $u_n = u_1 + (n-1)d$ where $u_n$ is the nth term, $u_1$ is the first term, and n is the position
• Sum formula: $S_n = \frac{n}{2}(2u_1 + (n-1)d)$ or $S_n = \frac{n}{2}(u_1 + u_n)$

Geometric Sequences: Each term is multiplied by a constant value called the common ratio (r).

• General term formula: $u_n = u_1 \cdot r^{n-1}$
• Sum formula (finite): $S_n = \frac{u_1(r^n - 1)}{r - 1}$ or $S_n = \frac{u_1(1 - r^n)}{1 - r}$ for $r \neq 1$
• Sum to infinity: $S_\infty = \frac{u_1}{1-r}$ when $|r| < 1$ (converging sequence)

Key Insight: Applications involve modelling real phenomena like loan repayments, population growth, depreciation, and compound interest.

Sigma Notation: $\sum_{k=1}^{n} u_k$ represents the sum of terms from $k=1$ to $k=n$

Recursive formulas define terms using previous terms: $u_{n+1} = u_n + d$ (arithmetic) or $u_{n+1} = r \cdot u_n$ (geometric)

Understanding when to apply each type and recognizing patterns in context is essential for Cambridge IB success.

Sequences model countless real-world phenomena, making them powerful problem-solving tools.

Arithmetic Sequences in Practice: Loan Repayments (Linear Depreciation): A car valued at $25,000 depreciates by $2,000 yearly forms an arithmetic sequence: 25000, 23000, 21000... The common difference d = -2000. After n years: $V_n = 25000 - 2000(n-1)$. This helps buyers predict resale value.

Seating Arrangements: A theatre has 20 seats in row 1, with each subsequent row having 3 more seats. Row n has $u_n = 20 + 3(n-1)$ seats. To find total seating for 15 rows, use $S_{15} = \frac{15}{2}(2(20) + 14(3)) = 615$ seats.

Geometric Sequences in Action: Population Growth: Bacteria doubling every hour: 100, 200, 400, 800... forms a geometric sequence with r = 2. After n hours: $P_n = 100 \cdot 2^{n-1}$. This exponential growth model applies to epidemiology and environmental science.

Compound Interest: Investing $5,000 at 6% annual interest compounded yearly creates the sequence: 5000, 5300, 5618... where $A_n = 5000(1.06)^{n}$. Understanding this empowers financial literacy.

Medicine Dosage: A drug with 30% daily elimination rate leaves 70% in the body. Starting with 200mg: 200, 140, 98... This geometric sequence (r = 0.7) helps determine safe dosing intervals.

Analogy: Think of arithmetic sequences as climbing stairs at constant steps, while geometric sequences are like climbing where each step is proportionally larger—like compound growth snowballing.

Recognizing which sequence type fits each scenario is crucial for accurate modelling.

Example 1: Savings Plan (Arithmetic Application)

Sarah deposits $150 monthly into a savings account (no interest). How much will she have after 3 years?

Solution: Step 1: Identify sequence type—constant monthly addition suggests arithmetic sequence

• $u_1 = 150$, d = 150 (each term is cumulative deposits)
• Number of terms: n = 3 × 12 = 36 months

Step 2: Apply sum formula: $S_n = \frac{n}{2}(2u_1 + (n-1)d)$

Step 3: Calculate: $S_{36} = \frac{36}{2}(2(150) + 35(150)) = 18(300 + 5250) = 18(5550) = 99,900$

Examiner Note: Always verify units match the question (months vs years)

Answer: $99,900 or Sarah saves $5,550 total

Example 2: Viral Video Views (Geometric Application)

A video gets 500 views on day 1, tripling daily. Find: (a) Views on day 7 (b) Total views in first week

Solution: Part (a):

• Geometric sequence: $u_1 = 500$, r = 3, n = 7
• $u_7 = 500 \cdot 3^{7-1} = 500 \cdot 3^6 = 500 \cdot 729 = 364,500$ views

Part (b):

• $S_7 = \frac{500(3^7 - 1)}{3-1} = \frac{500(2187-1)}{2} = \frac{500(2186)}{2} = 546,500$ total views

Examiner Note: Use correct formula format—calculator permitted but show method

Example 3: Convergent Series

Find sum to infinity: 12 + 6 + 3 + 1.5 +...

• $u_1 = 12$, r = 0.5 (since $|r| < 1$, converges)
• $S_\infty = \frac{12}{1-0.5} = \frac{12}{0.5} = 24$