Sequences and Series

Subject: Mathematics
Topic: 5
Cambridge Code: 0580

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Sequences

Sequence - Ordered list of numbers following pattern

Notation

General term: $u_n$ or $a_n$ Sequence: $u_1, u_2, u_3, ..., u_n$

Arithmetic Sequences

Arithmetic sequence - Constant difference between terms

Definition: $u_n = u_{n-1} + d$

Where d is common difference

General term: $u_n = a + (n-1)d$

Where a is first term

Example:

• Sequence: 3, 7, 11, 15, ...
• a = 3, d = 4
• $u_n = 3 + (n-1)4 = 4n - 1$

Geometric Sequences

Geometric sequence - Constant ratio between terms

Definition: $u_n = u_{n-1} \cdot r$

Where r is common ratio

General term: $u_n = ar^{n-1}$

Where a is first term

Example:

• Sequence: 2, 6, 18, 54, ...
• a = 2, r = 3
• $u_n = 2 \cdot 3^{n-1}$

Other Sequences

Fibonacci-type:

• $u_n = u_{n-1} + u_{n-2}$
• Example: 1, 1, 2, 3, 5, 8, ...

Quadratic:

• General term is polynomial
• Differences of differences constant

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Series

Series - Sum of sequence terms

$S_n = u_1 + u_2 + u_3 + ... + u_n = \sum_{i=1}^{n} u_i$

Arithmetic Series

Sum of arithmetic series: $S_n = \frac{n}{2}(2a + (n-1)d)$

Or equivalently: $S_n = \frac{n}{2}(a + l)$

Where l is last term ($l = a + (n-1)d$)

Example:

• Sum of 1 + 2 + 3 + ... + 100
• $S_{100} = \frac{100}{2}(1 + 100) = 50 \times 101 = 5050$

Geometric Series

Sum of geometric series (r ≠ 1): $S_n = a\frac{1 - r^n}{1 - r} = a\frac{r^n - 1}{r - 1}$

Example:

• Sum of 2 + 4 + 8 + 16 (4 terms)
• a = 2, r = 2, n = 4
• $S_4 = 2\frac{2^4 - 1}{2 - 1} = 2(15) = 30$

Infinite Series

Convergent series:

• Sum approaches finite limit as n → ∞
• Possible only if |r| < 1 (geometric)

Sum to infinity (geometric, |r| < 1): $S_∞ = \frac{a}{1 - r}$

Example:

• Infinite series: $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...$
• a = 1, r = 1/2
• $S_∞ = \frac{1}{1 - 1/2} = 2$

Divergent series:

• Sum does not converge
• |r| ≥ 1 (geometric)

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Summation Notation

$\sum_{i=1}^{n} u_i = u_1 + u_2 + ... + u_n$

Properties

$\sum_{i=1}^{n} c \cdot u_i = c \sum_{i=1}^{n} u_i$

$\sum_{i=1}^{n} (u_i + v_i) = \sum_{i=1}^{n} u_i + \sum_{i=1}^{n} v_i$

Standard Sums

$\sum_{i=1}^{n} c = cn$

$\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$

$\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$

$\sum_{i=1}^{n} i^3 = \left(\frac{n(n+1)}{2}\right)^2$

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Binomial Theorem

Binomial expansion: $(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k$

Where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$

Example

$(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3$

Binomial Coefficients

Pascal's Triangle:

        1
       1 1
      1 2 1
     1 3 3 1
    1 4 6 4 1

Each entry is sum of two above

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Applications

Growth and Decay

Exponential growth: $A = A_0 \cdot r^t$

• Population, bacteria, investments
• r > 1: growth
• 0 < r < 1: decay

Annuities

Future value (regular deposits): $FV = PMT \cdot \frac{(1 + i)^n - 1}{i}$

Where PMT is payment, i is interest rate per period

Compound Interest

$A = P(1 + r)^n$

Where P is principal, r is rate, n is periods

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Testing Convergence

Geometric Series Test

Convergent if: |r| < 1

Ratio Test

$L = \lim_{n \to ∞} \left|\frac{u_{n+1}}{u_n}\right|$

• L < 1: Convergent
• L > 1: Divergent
• L = 1: Inconclusive

Divergence Test

If $\lim_{n \to ∞} u_n ≠ 0$, then series diverges

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Key Points

1. Arithmetic sequence has constant difference
2. Geometric sequence has constant ratio
3. $u_n = a + (n-1)d$ for arithmetic
4. $u_n = ar^{n-1}$ for geometric
5. Sum of arithmetic: $S_n = \frac{n}{2}(a+l)$
6. Sum of geometric: $S_n = a\frac{1-r^n}{1-r}$
7. Geometric series converges if |r| < 1
8. Sum to infinity: $S_∞ = \frac{a}{1-r}$ (|r| < 1)
9. Binomial theorem expands (a+b)^n
10. Understanding patterns is key

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Practice Questions

1. Find general term of sequences
2. Calculate specific terms
3. Find sum of arithmetic series
4. Find sum of geometric series
5. Determine convergence of series
6. Find sum to infinity
7. Solve growth/decay problems
8. Apply binomial theorem
9. Use summation notation
10. Model real-world situations

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Revision Tips

• Distinguish arithmetic vs geometric quickly
• Memorize key formulas
• Understand convergence conditions
• Practice finding general terms
• Work many series problems
• Connect to real applications
• Verify answers reasonably
• Use summation notation correctly