Sequences & Series

Example:

3, 7, 11, 15, 19, ... (common difference d = 4)

20, 15, 10, 5, 0, ... (common difference d = -5)

\[ u_n = u_1 + (n-1)d \]

Where:

• • \( u_n \) = nth term (the term you want to find)
• • \( u_1 \) = first term
• • \( n \) = term number (position in the sequence)
• • \( d \) = common difference

💡 How to Find the Common Difference:

\[ d = u_2 - u_1 = u_3 - u_2 = u_{n+1} - u_n \]

Step 1: Identify the given values

\( u_1 = 5 \)

\( d = 9 - 5 = 4 \)

\( n = 20 \)

Step 2: Apply the formula

\( u_{20} = 5 + (20-1) \times 4 \)

\( u_{20} = 5 + 19 \times 4 \)

\( u_{20} = 5 + 76 \)

\( u_{20} = 81 \)

✓ The 20th term is 81

Formula 1 (when you know first and last terms):

\[ S_n = \frac{n}{2}(u_1 + u_n) \]

Formula 2 (when you know first term and common difference):

\[ S_n = \frac{n}{2}[2u_1 + (n-1)d] \]

Where:

• • \( S_n \) = sum of the first n terms
• • \( n \) = number of terms
• • \( u_1 \) = first term
• • \( u_n \) = nth term (last term)
• • \( d \) = common difference

Step 1: Identify the given values

\( u_1 = 3 \), \( d = 4 \), \( n = 15 \)

Step 2: Choose the appropriate formula (Formula 2)

\( S_{15} = \frac{15}{2}[2(3) + (15-1)(4)] \)

\( S_{15} = \frac{15}{2}[6 + 14 \times 4] \)

\( S_{15} = \frac{15}{2}[6 + 56] \)

\( S_{15} = \frac{15}{2} \times 62 \)

\( S_{15} = 15 \times 31 = 465 \)

✓ The sum of the first 15 terms is 465

Example:

2, 6, 18, 54, 162, ... (common ratio r = 3)

80, 40, 20, 10, 5, ... (common ratio r = 0.5)

3, -6, 12, -24, 48, ... (common ratio r = -2)

\[ u_n = u_1 \cdot r^{n-1} \]

Where:

• • \( u_n \) = nth term
• • \( u_1 \) = first term
• • \( r \) = common ratio
• • \( n \) = term number

💡 How to Find the Common Ratio:

\[ r = \frac{u_2}{u_1} = \frac{u_3}{u_2} = \frac{u_{n+1}}{u_n} \]

Step 1: Identify the given values

\( u_1 = 2 \)

\( r = \frac{6}{2} = 3 \)

\( n = 8 \)

Step 2: Apply the formula

\( u_8 = 2 \times 3^{8-1} \)

\( u_8 = 2 \times 3^7 \)

\( u_8 = 2 \times 2187 \)

\( u_8 = 4374 \)

✓ The 8th term is 4374

Formula (most common form):

\[ S_n = \frac{u_1(r^n - 1)}{r - 1} \quad \text{when } r > 1 \]

Alternative form:

\[ S_n = \frac{u_1(1 - r^n)}{1 - r} \quad \text{when } r < 1 \]

💡 Both formulas are equivalent! Choose whichever is easier for your calculation.

\[ S_\infty = \frac{u_1}{1 - r} \]

Convergence Rules:

• If \( |r| < 1 \): Series converges (has a finite sum)
• If \( |r| \geq 1 \): Series diverges (sum approaches infinity)

Step 1: Identify the given values

\( u_1 = 3 \), \( r = \frac{6}{3} = 2 \), \( n = 6 \)

Step 2: Apply the formula (r > 1)

\( S_6 = \frac{3(2^6 - 1)}{2 - 1} \)

\( S_6 = \frac{3(64 - 1)}{1} \)

\( S_6 = 3 \times 63 = 189 \)

✓ The sum is 189

\[ \sum_{r=a}^{b} f(r) \]

Where:

• • Σ = sum symbol
• • r = index of summation (can be any variable)
• • a = lower limit (starting value)
• • b = upper limit (ending value)
• • f(r) = expression to be summed

Example 1: Evaluate \( \displaystyle\sum_{r=1}^{5} (2r + 3) \)

Example 2: Evaluate \( \displaystyle\sum_{k=3}^{6} k^2 \)

ARITHMETIC

GEOMETRIC

💡 Tip 1: Use Your GDC (Calculator) Wisely

Your calculator can evaluate sigma notation and check sums! Use it to verify answers, but always show your working for full marks.

Calculator functions: Look for "sum(" or "Σ" in your GDC menu.

💡 Tip 2: Know Which Formula to Use

Arithmetic sum: Use Formula 1 if you know first and last terms; use Formula 2 if you know common difference.
Geometric sum: Check if you need finite or infinite sum formula.

💡 Tip 3: Identify the Sequence Type First

Check consecutive terms: If you add/subtract the same value → Arithmetic. If you multiply/divide by the same value → Geometric.

💡 Tip 4: Watch for Simultaneous Equations

IB often gives you two pieces of information (e.g., "3rd term is 10 and 7th term is 22"). Set up two equations with \( u_1 \) and \( d \) (or \( r \)), then solve simultaneously.

💡 Tip 5: Check Convergence for Infinite Series

Before using \( S_\infty = \frac{u_1}{1-r} \), always verify that \( |r| < 1 \). If \( |r| \geq 1 \), the series diverges and has no finite sum.

💡 Tip 6: Be Careful with Sigma Notation Limits

Not all sigma notation starts at r = 1! Always check the lower limit. If it starts at r = 3, begin your sum from there, not from 1.

💡 Tip 7: Application Problems

Finance: Compound interest uses geometric sequences.
Linear patterns: Saving the same amount weekly uses arithmetic sequences.
Population growth: Often modeled with geometric sequences.

❌ Wrong: "Find the 5th term" → Answer: 5

✓ Correct: "Find the 5th term" → Answer: \( u_5 \) = (calculated value)

❌ Wrong: \( u_n = u_1 \cdot r^n \)

✓ Correct: \( u_n = u_1 \cdot r^{n-1} \)

If r = 2, the series diverges — no finite sum exists!

Sequence: 2, 4, 8, 16, ...

❌ Arithmetic? 4 - 2 = 2, but 8 - 4 = 4 (not constant!)

✓ Geometric! 4/2 = 2, 8/4 = 2, 16/8 = 2 (constant ratio!)