Sequences and Series

Q: Do I need to memorise the AP and GP sum formulae?

While these formulae are provided in the exam formula booklet, you must be able to derive both proofs explicitly, as derivation questions are regularly examined.

Q: When can I use the sum to infinity formula for a GP?

The sum to infinity only applies if the common ratio has an absolute value less than 1 (), meaning the series converges to a fixed finite value.

Q: Is binomial expansion for negative exponents covered in P2?

No, P2 only requires binomial expansion for positive integer powers of . Negative and rational exponents are covered in Pure Mathematics 4 (P4).

Edexcel International A-Level Mathematics· 2018 Specification (Issue 3), first assessment Jan 2019· 35 min read

1. Types of Sequences and Recurrence Relations★★☆☆☆⏱ 8 min

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📘 Definition

Sequence

An ordered list of terms defined by either an nth term formula $u_{n} = f (n)$ or a recurrence relation $u_{n + 1} = f (u_{n})$ with a given initial term $u_{1}$ .

Example:

The sequence of even numbers has nth term $u_{n} = 2 n$ .

Sequences are classified as:
• Increasing: $u_{n + 1} > u_{n}$ for all $n \in N$
• Decreasing: $u_{n + 1} < u_{n}$ for all $n \in N$
• Periodic: $u_{n + k} = u_{n}$ for all $n$ , where $k$ is the period (cycle length).

📐 Worked Example

A sequence is defined by $u_{1} = 3$ , $u_{n + 1} = 2 u_{n} - 1$ . Find the first 4 terms, classify the sequence, and state its period if periodic.

1
Calculate first 4 terms: $u_{1} = 3$ , $u_{2} = 2 (3) - 1 = 5$ , $u_{3} = 2 (5) - 1 = 9$ , $u_{4} = 2 (9) - 1 = 17$
2
Check classification: Each term is larger than the previous, so this is an increasing sequence.
3
Check periodicity: Terms do not repeat, so the sequence is not periodic.

Exam tip:

Recurrence relation questions often ask you to confirm sequence convergence; for Edexcel P2, only apply the GP convergence rule $∣ r ∣ < 1$ for geometric recurrence relations.

2. Arithmetic Sequences and Series (AP)★★★☆☆⏱ 10 min

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📘 Definition

Arithmetic Progression (AP)

$u_{n} = a + (n - 1) d$ , $S_{n} = \frac{1}{2} n [2 a + (n - 1) d]$

Sequence with constant common difference $d$ between consecutive terms, where $a$ is the first term.

Example:

The sequence 2, 5, 8, 11 has $a = 2$ , $d = 3$ .

🔬 Derivation

Goal:

Prove the sum of first n terms of an AP

1
Write the sum forwards: $S_{n} = a + (a + d) + (a + 2 d) + ... + (a + (n - 1) d)$
2
Write the sum backwards: $S_{n} = (a + (n - 1) d) + (a + (n - 2) d) + ... + a$
3
Add both equations: $2 S_{n} = n [2 a + (n - 1) d]$
4
Rearrange: $S_{n} = \frac{1}{2} n [2 a + (n - 1) d] = \frac{1}{2} n (a + l)$ , where $l$ is the last term.

Result:

The sum formula is valid for all finite arithmetic progressions.

The sum of the first $n$ natural numbers is a special case of AP with $a = 1$ , $d = 1$ : $S_{n} = \frac{1}{2} n (n + 1)$ . Sigma notation $\sum_{r = 1}^{n} u_{r}$ is used to denote the sum of terms from $r = 1$ to $r = n$ .

📐 Worked Example

Find the sum of the first 20 terms of the AP 7, 11, 15, 19... Verify your result using the AP sum proof structure.

1
Identify values: $a = 7$ , $d = 4$ , $n = 20$
2
Calculate 20th term: $u_{20} = 7 + (20 - 1) (4) = 83$
3
Use sum formula: $S_{20} = \frac{1}{2} * 20 * (7 + 83) = 10 * 90 = 900$
4
Verification: Forwards + backwards sum gives 20 pairs each summing to 90, so $2 S_{20} = 20 * 90 = 1800$ , confirming $S_{20} = 900$ .

Exam tip:

Proof of the AP sum formula is a 3-4 mark question that appears regularly; memorise the forward/backward addition method exactly.

3. Geometric Sequences and Series (GP)★★★☆☆⏱ 10 min

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📘 Definition

Geometric Progression (GP)

$u_{n} = a r^{n - 1}$ , $S_{n} = \frac{a ( 1 - r ^{n} )}{1 - r}$ ( $r \neq = 1$ ), $S_{\infty} = \frac{a}{1 - r}$ ( $∣ r ∣ < 1$ )

Sequence with constant common ratio $r$ between consecutive terms, where $a$ is the first term.

Example:

The sequence 2, 6, 18, 54 has $a = 2$ , $r = 3$ .

🔬 Derivation

Goal:

Prove the sum of first n terms of a GP

1
Write the sum: $S_{n} = a + a r + a r^{2} + ... + a r^{n - 1}$
2
Multiply both sides by $r$ : $r S_{n} = a r + a r^{2} + ... + a r^{n - 1} + a r^{n}$
3
Subtract the second equation from the first: $S_{n} - r S_{n} = a - a r^{n}$
4
Factor and rearrange: $S_{n} (1 - r) = a (1 - r^{n})$ → $S_{n} = \frac{a ( 1 - r ^{n} )}{1 - r}$ for $r \neq = 1$ .

Result:

The sum formula is valid for all finite geometric progressions with $r \neq = 1$ .

For convergent GPs where $∣ r ∣ < 1$ , as $n \to \infty$ , $r^{n} \to 0$ , so the sum to infinity is $S_{\infty} = \frac{a}{1 - r}$ . Use logarithms to solve for $n$ when given a target sum or term value.

📐 Worked Example

A GP has first term 100 and common ratio 0.8. Find: (a) the sum of the first 10 terms, (b) the sum to infinity, (c) the smallest value of $n$ for which $u_{n} < 1$ .

1
(a) Sum of first 10 terms: $S_{10} = \frac{100 ( 1 - 0. 8 ^{10} )}{1 - 0.8} = 500 (1 - 0.1074) = 446.3$ (1 dp)
2
(b) $∣0.8∣ < 1$ , so sum to infinity: $S_{\infty} = \frac{100}{1 - 0.8} = 500$
3
(c) Set $u_{n} < 1$ : $100 * 0. 8^{n - 1} < 1$ → $0. 8^{n - 1} < 0.01$ . Take natural logs of both sides: $(n - 1) ln (0.8) < ln (0.01)$ . $ln (0.8)$ is negative, so reverse inequality: $n - 1 > \frac{l n ( 0.01 )}{l n ( 0.8 )} \approx 20.6$ , so smallest integer $n = 22$ .

Exam tip:

Always explicitly state that $∣ r ∣ < 1$ when using the sum to infinity formula, as this is a required mark point in Edexcel exams.

4. Binomial Expansion for Positive Integer n★★★★☆⏱ 7 min

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📘 Definition

Binomial Expansion (Positive Integer n)

$(a + b)^{n} = \sum_{r = 0}^{n} (r n) a^{n - r} b^{r}$ , $(r n) =^{n} C_{r} = \frac{n !}{r ! ( n - r )!}$

Expansion of a binomial raised to a positive integer power $n$ , where $^{n} C_{r}$ is the binomial coefficient.

Example:

$(1 + x)^{3} = 1 + 3 x + 3 x^{2} + x^{3}$

For expansions of the form $(a + b x)^{n}$ , substitute $b$ with $b x$ in the standard formula. You can use the $^{n} C_{r}$ function on your calculator to compute binomial coefficients quickly, but you must show your working for method marks.

📐 Worked Example

Find the first 4 terms of the expansion of $(2 - 3 x)^{5}$ in ascending powers of $x$ , and state the coefficient of $x^{3}$ .

1
Identify values: $a = 2$ , $b = - 3 x$ , $n = 5$
2
$\binom{5}{0}2^5(-3x)^0 = 1 * 32 * 1 = 32$
3
$\binom{5}{1}2^4(-3x)^1 = 5 * 16 * (-3x) = -240x$
4
$\binom{5}{2}2^3(-3x)^2 = 10 * 8 * 9x^2 = 720x^2$
5
$\binom{5}{3}2^2(-3x)^3 = 10 * 4 * (-27x^3) = -1080x^3$
6
First 4 terms: $32 - 240 x + 720 x^{2} - 1080 x^{3}$ . Coefficient of $x^{3}$ is -1080.

Exam tip:

Always apply the power to the entire coefficient of x in binomial expansions (including the sign) to avoid common sign errors.

5. Common Pitfalls

Wrong move:

Forgetting to reverse the inequality sign when dividing by the log of a number <1 when solving for n in GP problems.

Why:

Logarithms of numbers between 0 and 1 are negative, so dividing by a negative value flips the inequality direction.

Correct move:

Check the sign of the log value before dividing, and explicitly reverse the inequality if the divisor is negative.

Wrong move:

Using the sum to infinity formula for a GP with |r| ≥1.

Why:

If |r| ≥1, the terms do not approach 0, so the series diverges and has no finite sum to infinity.

Correct move:

Always verify |r|<1 before applying the sum to infinity formula, and state this condition explicitly in exam answers.

Wrong move:

In the AP sum proof, adding the forward and backward sums incorrectly.

Why:

Each pair of terms sums to the same value (a + l, where l is the last term), so there are n identical pairs, giving 2S_n = n(a+l).

Correct move:

Memorise the full proof structure: write S_n forwards, then backwards, add term by term, rearrange for S_n.

Wrong move:

Forgetting to apply the power to the entire coefficient of x in binomial expansions.

Why:

If the term is (-3x)^r, the power applies to both the -3 and the x, not just the x.

Correct move:

Put brackets around the entire x term when expanding to ensure you apply the power to the coefficient and sign as well as the variable.

Wrong move:

Confusing nth term formulae for AP and GP: using ar^{n-1} for AP or a+(n-1)d for GP.

Why:

AP uses additive common difference, GP uses multiplicative common ratio, so formulae are not interchangeable.

Correct move:

Check if the sequence has constant difference (AP) or constant ratio (GP) before selecting the nth term formula.

6. Quick Reference Cheatsheet

Concept	Formula	Key Notes	Exam Requirement
Sequence Classification	$u_{n + 1} > u_{n}$ (increasing), $u_{n + 1} < u_{n}$ (decreasing), $u_{n + k} = u_{n}$ (period k)	Test ≥4 terms to confirm pattern	Classify given sequences/recurrence relations
AP nth term	$u_{n} = a + (n - 1) d$	a = first term, d = common difference	Solve for unknown terms/difference
AP Sum	$S_{n} = \frac{1}{2} n (2 a + (n - 1) d) = \frac{1}{2} n (a + l)$	l = last term	Proof required; calculate finite sums
GP nth term	$u_{n} = a r^{n - 1}$	a = first term, r = common ratio	Use logs to solve for n
GP Sum	$S_{n} = \frac{a ( 1 - r ^{n} )}{1 - r}$ (r≠1), $S_{\infty} = \frac{a}{1 - r}$ (\|r\|<1)	Only use S∞ for convergent series	Proof required; state \|r\|<1 condition for S∞
Binomial Expansion	$(a + b x)^{n} = \sum_{r = 0}^{n} (r n) a^{n - r} (b x)^{r}$	$(r n) =^{n} C_{r} = \frac{n !}{r ! ( n - r )!}$ , n positive integer only	Find specific coefficients/terms

7. Frequently Asked

Do I need to memorise the AP and GP sum formulae?

While these formulae are provided in the exam formula booklet, you must be able to derive both proofs explicitly, as derivation questions are regularly examined.

When can I use the sum to infinity formula for a GP?

The sum to infinity $S_{\infty} = \frac{a}{1 - r}$ only applies if the common ratio has an absolute value less than 1 ( $∣ r ∣ < 1$ ), meaning the series converges to a fixed finite value.

Is binomial expansion for negative exponents covered in P2?

No, P2 only requires binomial expansion for positive integer powers of $(a + b x)$ . Negative and rational exponents are covered in Pure Mathematics 4 (P4).

Going deeper

What's Next

Now that you have mastered P2 sequences and series, you are ready to move on to more advanced pure mathematics topics for your IAL exams. Next, you will study differentiation and integration for trigonometric functions, which builds on your algebra and series knowledge to solve calculus problems. You can also practice exam-style questions for this topic to reinforce your understanding, and review proof techniques that will be used in later topics like P4 binomial series for rational exponents and further calculus. Make sure you memorise the required AP and GP sum proofs, as these are high-frequency low-difficulty marks that you should not lose in your exam.