Sequences and series — Edexcel A-Level Mathematics
Test yourself on Sequences and series with PEARSON EDEXCEL A-Level practice questions.
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Sequences and series explained
The binomial expansion writes (a + bx)ⁿ as a sum of terms.
Read the full explanation
For positive integer n the series terminates after n + 1 terms, and coefficients come from n! and ⁿCᵣ, where n! = n × (n − 1) × ... × 1 and ⁿCᵣ = n! ÷ [r!(n − r)!]. These same coefficients count combinations in binomial probability. For rational n, including negative and fractional values, the expansion continues indefinitely and is valid only when |bx/a| < 1. For example, (1 + 2x)⁻¹ expands to 1 − 2x + 4x² − ... valid for |x| < 1/2. Approximation uses the first few terms when x is small, and the validity condition ensures the infinite series converges.
4.2 Work with sequences including those given by a formula for the nth term and those generated by a simple relation of the form xₙ₊₁ = f(xₙ); increasing sequences; decreasing sequences; periodic sequences.
A sequence is an ordered list of terms. Some sequences have an explicit formula for the nth term, such as uₙ = 3n + 1, so any term can be found directly. Others are defined recursively by a relation like xₙ₊₁ = f(xₙ), where each term depends on the previous one; these need a starting value. Sequences are classified by behaviour: increasing if every term is greater than the one before, decreasing if every term is smaller, and periodic if a block of terms repeats, with the period being the length of that block. For example, xₙ₊₁ = xₙ + 4 with x₁ = 1 gives 1, 5, 9, 13, an increasing sequence. Checking differences or ratios between consecutive terms identifies the type.
4.3 Understand and use sigma notation for sums of series.
Sigma notation compactly writes a sum. In Σ from r = 1 to n of uᵣ, the letter r is the index, 1 is the lower limit, n is the upper limit, and uᵣ is the general term. To evaluate, substitute each integer index from the lower to the upper limit and add the results. For example, Σ from r = 1 to 4 of (2r + 1) gives 3 + 5 + 7 + 9 = 24. Sigma notation works with arithmetic and geometric series, and standard results such as Σr = n(n + 1)/2 and Σr² = n(n + 1)(2n + 1)/6 allow sums to be found without listing every term. You must also be able to rewrite an expanded sum in sigma form and adjust limits when the index starts at a value other than 1.
4.4 Understand and work with arithmetic sequences and series, including the formulae for nth term and the sum to n terms.
An arithmetic sequence adds a constant common difference d to each term, so a, a + d, a + 2d, ... The nth term is uₙ = a + (n − 1)d, where a is the first term. The sum of the first n terms is Sₙ = n/2[2a + (n − 1)d], equivalently Sₙ = n/2(a + l) when the last term l is known. For example, 5, 8, 11, 14, ... has a = 5 and d = 3, so u₁₀ = 5 + 9 × 3 = 32 and S₁₀ = 10/2(2 × 5 + 9 × 3) = 5 × 37 = 185. To work with these, identify a and d, substitute into the correct formula, and solve the resulting linear or quadratic equation. Recognise that d may be negative or a fraction, and that n must be a positive integer.
4.5 Understand and work with geometric sequences and series, including the formulae for the nth term and the sum of a finite geometric series; the sum to infinity of a convergent geometric series, including the use of |r| < 1; modulus notation.
A geometric sequence multiplies by a constant common ratio r each time, so a, ar, ar², ar³, ... The nth term is uₙ = ar^(n − 1). The sum of the first n terms is Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1, or equivalently a(rⁿ − 1)/(r − 1). When |r| < 1 the terms shrink towards zero and the series converges to S∞ = a/(1 − r); the modulus notation |r| < 1 means r lies strictly between −1 and 1. For example, 8, 4, 2, 1, ... has a = 8 and r = ½, so u₆ = 8 × (½)⁵ = ¼ and S∞ = 8/(1 − ½) = 16. Identify a and r, check convergence before using S∞, and solve equations involving powers, often using logarithms.
4.6 Use sequences and series in modelling.
Modelling translates a real situation into a sequence (individual terms) or series (cumulative totals). Decide if the pattern is arithmetic (constant difference) or geometric (constant ratio), define n, and state units. For example, a salary starting at £24 000 (n=1) with a £900 annual rise is arithmetic (a = 24 000, d = 900). The 10th year salary (sequence) is 24 000 + 9(900) = £32 100. The total earned over 10 years (series) is (10/2)[2(24 000) + 9(900)] = 5[48 000 + 8 100] = £280 500. A car worth £18 000 losing 15% each year is geometric (r = 0.85); after 5 full years, its value is 18 000 × 0.85⁵ ≈ £7 986.70. Check if n starts at 0 or 1, and if the answer needs rounding.
Your focus
- Expand (a + bx)ⁿ for positive integer n and for rational n.
- Evaluate and interpret n! and ⁿCᵣ in algebraic and probability contexts.
- Determine and apply the validity condition |bx/a| < 1 when using an infinite expansion.
Show all 18 objectives
- Generate terms of a sequence from an nth-term formula or a recurrence relation.
- Classify sequences as increasing, decreasing or periodic with justification.
- Determine the period of a periodic sequence from its terms.
- Interpret and evaluate expressions written in sigma notation.
- Express a given series in sigma notation with correct limits and general term.
- Apply standard summation results to find sums of series.
- Calculate any term of an arithmetic sequence using uₙ = a + (n − 1)d.
- Calculate the sum of the first n terms using Sₙ = n/2[2a + (n − 1)d] or Sₙ = n/2(a + l).
- Determine a, d or n from given information and interpret the result in context.
- Calculate any term of a geometric sequence using uₙ = ar^(n − 1).
- Calculate the sum of a finite geometric series using Sₙ = a(1 − rⁿ)/(1 − r).
- Determine whether a geometric series converges using |r| < 1 and calculate its sum to infinity when it does.
- Translate a real situation into an arithmetic or geometric sequence or series.
- Solve problems involving growth, decay, savings and repeated percentage change.
- Distinguish between sequence terms and series sums in context.
Sequences and series exam tips
Marking Points
- Expands (a + bx)ⁿ for positive integer n using ⁿCᵣ coefficients and correct ascending powers of x.
- Evaluates n! and ⁿCᵣ correctly, including cases such as ⁵C₂ = 10 and 4! = 24.
- Links binomial coefficients to binomial probability by identifying ⁿCᵣ as the number of ways of choosing r successes from n trials.
- Extends the expansion to rational n, forming the general term with the product of descending factors divided by r!.
- States and applies the validity condition |bx/a| < 1, and uses a sufficient number of terms to approximate a value to a required accuracy.
- Generates terms from an nth-term formula by substituting n = 1, 2, 3, ... in order.
- Uses a recurrence relation xₙ₊₁ = f(xₙ) with a given initial term to produce successive terms accurately.
- Classifies a sequence as increasing, decreasing or periodic by comparing consecutive terms or identifying a repeating block.
- States the period of a periodic sequence by counting the terms in the repeating cycle.
- Distinguishes between explicit and recursive definitions and explains when each is more useful.
- Interprets each part of sigma notation correctly: index, lower limit, upper limit and general term.
- Evaluates a finite sum by substituting consecutive integer values of the index and adding the terms.
- Rewrites an expanded series using sigma notation with a correct general term and limits.
- Applies standard summation results such as Σr = n(n + 1)/2 and Σr² = n(n + 1)(2n + 1)/6.
- Handles sums whose index starts at a value other than 1 by adjusting the general term or subtracting a partial sum.
- States uₙ = a + (n − 1)d and substitutes the correct a, d and n to find a term or to form an equation.
- Uses Sₙ = n/2[2a + (n − 1)d] or Sₙ = n/2(a + l), choosing the form that matches the given information.
- Finds a and d from contextual or algebraic information, for example from two given terms or from simultaneous equations.
- Solves equations arising from uₙ or Sₙ, including quadratic equations in n, and rejects non-integer or non-positive values of n.
- Interprets the result in context, such as identifying which term number gives a stated value or the total after a given number of terms.
- States uₙ = ar^(n − 1) and substitutes the correct a, r and n to find a term or form an equation.
- Uses Sₙ = a(1 − rⁿ)/(1 − r) for a finite geometric series, including when r is negative or fractional.
- Applies the convergence condition |r| < 1 and calculates S∞ = a/(1 − r) only when that condition holds.
- Finds a or r from given terms, for example by dividing two terms to eliminate a, and handles negative or fractional ratios.
- Solves equations involving rⁿ, using logarithms where necessary, and interprets whether a series converges or diverges.
- Identifies whether a context is arithmetic or geometric from the wording, such as a fixed annual increase or a percentage change.
- Defines a, d or r and n with correct units and states what n represents, including whether the first term corresponds to n = 0 or n = 1.
- Forms and solves an equation or inequality from the model, for example finding when a value first exceeds a target.
- Distinguishes between finding a specific term (sequence) and a cumulative total (series), applying the correct formula.
- Interprets the solution in context, including rounding to a sensible degree of accuracy and rejecting values that do not fit the situation.
Examiner Tips
- 💡Write the general term first, then substitute r values, so the pattern and signs stay consistent.
- 💡State the validity condition in the same line as the expansion to avoid losing that mark.
- 💡For approximations, keep one more term than the required accuracy suggests, then round at the end.
- 💡List terms in a table with n and xₙ columns to keep the recurrence steps visible.
- 💡For periodic sequences, write out at least two full cycles before stating the period.
- 💡When asked to classify, give the comparison or repeating block as justification, not just the label.
- 💡Write out the first two or three terms and the last term before summing, to confirm the pattern.
- 💡When a sum starts at r = 0 or r = 2, adjust by adding or subtracting the missing terms explicitly.
- 💡Check standard-result formulas with a small value of n, such as n = 1, before trusting them in a long calculation.
- 💡Write down a and d explicitly before substituting, so arithmetic slips are easier to spot.
- 💡When two terms are given, form two equations and solve simultaneously rather than guessing a and d.
- 💡Check a sum by adding the first few terms manually or by using Sₙ = n/2(a + l) as a second method.
- 💡State the value of r and check |r| < 1 before quoting a sum to infinity.
- 💡When r is negative, keep the sign throughout and check whether terms alternate.
- 💡For equations in n, take logarithms of both sides and round up if the context requires a whole number of terms.
- 💡Underline the words that indicate the type of change, such as 'increases by £900' or 'decreases by 15% each year'.
- 💡Define your variables in words before substituting, and keep units with every value.
Common Mistakes
- Treating a rational-n expansion as finite and stopping after a few terms; instead continue the pattern and state that the series is infinite.
- Omitting the validity condition or writing it as |x| < 1 regardless of a and b; instead derive |bx/a| < 1 from the ratio of successive terms.
- Miscomputing ⁿCᵣ by cancelling factorials incorrectly; instead use ⁿCᵣ = n! ÷ [r!(n − r)!] and check with a small case.
- Starting a recurrence from the wrong initial value or using xₙ instead of xₙ₊₁; instead label the first term clearly and apply the relation step by step.
- Calling a sequence increasing when only some early terms rise; instead check the general comparison xₙ₊₁ > xₙ across the stated range.
- Miscounting the period by including the repeated term twice; instead identify the shortest block that repeats exactly.
- Substituting the upper limit only instead of every integer from the lower to the upper limit; instead list each index value in turn.
- Confusing the index letter with a variable in the general term and treating it as fixed; instead treat the index as the changing counter.
- Using Σr² = [n(n + 1)/2]² instead of n(n + 1)(2n + 1)/6; instead memorise each standard result separately and check with n = 1.
- Using uₙ = a + nd instead of a + (n − 1)d; correct by subtracting 1 from n before multiplying by d.
- Treating d as always positive; correct by calculating d = u₂ − u₁ and retaining its sign.
- Accepting a negative or fractional value of n from a quadratic; correct by rejecting it because n counts terms and must be a positive integer.
- Using uₙ = arⁿ instead of ar^(n − 1); correct by using the exponent n − 1.
- Applying S∞ = a/(1 − r) when |r| ≥ 1; correct by checking the modulus condition first and stating that the series diverges.
- Dividing terms in the wrong order when finding r; correct by dividing a later term by the previous term, uₙ₊₁ ÷ uₙ.
- Confusing a percentage increase with a common difference; correct by converting a percentage change into a multiplier, for example a 15% decrease gives r = 0.85.
- Miscounting the number of steps because n starts at 0 rather than 1; correct by writing out the first few terms to check the indexing and defining n explicitly.
- Calculating the nth term instead of the sum of n terms (or vice versa); correct by reading carefully whether the question asks for a value in a specific year or a total over a period.