TopicFurther Maths
Pure · SS1–SS2
Sequences, series & binomial expansion
Sequences and series, sums to infinity, and expanding (a + b)ⁿ with the binomial theorem.
- Before this
- Sequences & series (AP, GP)
Learn
- 1Harder arithmetic progressionsA.P. questions with two facts to combine, a number of terms found from a quadratic, the least number of terms to pass a total, word problems, and sequences defined by a rule.Done18 min
- 2Harder geometric progressionsG.P. questions with two facts (divide to remove a), three terms written as a/r, a, ar, the sum to infinity combined with another fact, and an A.P. and a G.P. in the same question.Done18 min
- 3The binomial expansionExpand (a + bx)ⁿ with Pascal's triangle or ⁿCᵣ, find one coefficient without the whole expansion, find unknowns from given coefficients, use the first few terms to approximate a power, and expand (1 + x)ⁿ for a fractional or negative n.Done24 min
Flashcards
13 cards: the rules, facts and methods to know by heart, with the diagrams from the lessons. They come back until you know them.
Practise
Answer sequences, series & binomial expansion questions one at a time. You'll know straight away whether you're right, with hints and the working step by step.
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Past questions (43)
From 2008–2023: 40 WAEC, 3 NECO. Theory questions first, then objective, newest first.
Theory (28)
- WAEC 2023 · Paper 2 · Q11Using a scale of 2 cm to 2 units on both axes, indicate on a graph sheet the area bounded by the inequalities , …
- WAEC 2022 · Paper 2 · Q11Find the binomial expansion of and .
- WAEC 2022 · Paper 2 · Q9Given that , and are the first 3 terms of a linear sequence (A.P.), find the:
- WAEC 2022 · Paper 2 · Q9Evaluate , leaving the answer in the form , where , and are real numbers.
- WAEC 2020 · Paper 2 · Q3The first and second terms of a linear sequence (A.P.) are and respectively. Determine the least number of terms of the A.P. that …
- WAEC 2019 · Paper 2 · Q10Find the range of values of for which has real roots.
- WAEC 2019 · Paper 2 · Q3How many terms of the series add up to 165?
- WAEC 2019 · Paper 2 · Q2The fourth and sixth terms of a Geometric Progression (G.P.) are 54 and 486 respectively. If , find the third term.
- WAEC 2018 · Paper 2 · Q11(i) Write down the binomial expansion of in ascending powers of . (ii) Using the expansion in (a)(i), find, …
- WAEC 2018 · Paper 2 · Q4The sum of the first twelve terms of an Arithmetic Progression is 168. If the third term is 7, find the values of the common difference and the …
- WAEC 2018 · Paper 2 · Q3The sum of the second and third terms of a Geometric Progression (G.P.) is 48. If the sum of the third and fourth terms is 144, find the first …
- WAEC 2018 · Paper 2 · Q9The radius of a sphere increased by . Find the percentage increase in the volume.
- WAEC 2017 · Paper 2 · Q11If and , find .
- WAEC 2017 · Paper 2 · Q2The th term of an Arithmetic Progression (A.P.) is given by . Find the sum of the first terms of the Arithmetic Progression …
- WAEC 2016 · Paper 2 · Q10(i) Write down the expansion of in ascending powers of . (ii) If the coefficients of the fifth, sixth and seventh terms in the …
- WAEC 2014 · Paper 2 · Q11The sum to infinity of an exponential sequence (G.P.) with a positive common ratio is 25 and the sum of the first two terms is 16. Find the:
- WAEC 2013 · Paper 2 · Q1If the coefficients of and in the expansion of are equal, express in terms of .
- WAEC 2013 · Paper 2 · Q11The sum of the first three terms of a decreasing exponential sequence (G.P.) is equal to 7 and the product of these three terms is equal to 8. …
- WAEC 2013 · Paper 2 · Q4Find the sum of all the odd numbers from 51 to 99.
- WAEC 2012 · Paper 2 · Q9The first three terms of the expansion of in ascending powers of are . Find the values of and .
- WAEC 2011 · Paper 2 · Q3Write down the binomial expansion of in ascending powers of .
- WAEC 2011 · Paper 2 · Q4Find the third term of the exponential sequence (GP)
- WAEC 2010 · Paper 2 · Q2If the sum of the first terms of the series is , find .
- WAEC 2009 · Paper 2 · Q9The 3rd and 6th terms of a geometric progression (G.P.) are 2 and 54 respectively. Find the: (i) common ratio; (ii) first term; (iii) sum of …
- WAEC 2009 · Paper 2 · Q10An exponential sequence is given by Find an expression for the th term;
- WAEC 2009 · Paper 2 · Q4Write down the first three terms of the binomial expansion of in ascending powers of .
- WAEC 2008 · Paper 2 · Q9The sum of the first terms of a sequence is given by . Write down the first four terms of the sequence …
- WAEC 2008 · Paper 2 · Q9A curve has gradient at the point . If it passes through the point , find its equation.
Objective (15)
- NECO 2023 · Paper 1 · Q18Find the first 4 terms in the expansion of .
- NECO 2023 · Paper 1 · Q5Find the sum of the first 9 terms of the exponential sequence
- NECO 2023 · Paper 1 · Q9If and , , determine the fifth term of the sequence.
- WAEC 2023 · Paper 1 · Q10An exponential sequence (G.P.) is given by . Find its sum to infinity.
- WAEC 2023 · Paper 1 · Q23Find the fifth term in the binomial expansion of .
- WAEC 2023 · Paper 1 · Q37Find the coefficient of the term in the binomial expansion of in ascending powers of .
- WAEC 2023 · Paper 1 · Q38An exponential sequence (G.P.) is given by . Find the term of the sequence.
- WAEC 2022 · Paper 1 · Q14Find the coefficient of in the binomial expansion of .
- WAEC 2022 · Paper 1 · Q17If , , , find the values of and .
- WAEC 2022 · Paper 1 · Q25Find the coefficient of in the binomial expansion of .
- WAEC 2022 · Paper 1 · Q26The first, second and third terms of an exponential sequence (G.P.) are , and respectively. Find the values of …
- WAEC 2020 · Paper 1 · Q13Determine the coefficient of in the binomial expansion of .
- WAEC 2020 · Paper 1 · Q21In which of the following series can the formula , where is the first term and is the common ratio, be used to …
- WAEC 2020 · Paper 1 · Q22If the binomial expansion of is used to evaluate , find the value of .
- WAEC 2020 · Paper 1 · Q23Find the th term of the linear sequence (A.P.)