You should be able to identify arithmetic and geometric sequences, and be able to generate the nth term formula as well as the ‘term-to-term’ or recurrence formula for a variety of sequences. You should be confident with the ‘common difference’ a and ‘common ratio’ r and be able to use these to set up equations based on the relationship between different (not necessarily consecutive) terms of a sequence.

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You should understand the concept of convergence for sequences and geometric series and the restrictions on the common ratio required to give a convergent geometric series. Note also that if a sequence converges to a limit L then applying the recurrence formula to that limit will also give each subsequent term to be L.

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As there is no formula booklet given for the TMUA exam, you will need to memorise certain formulae that you might usually look up. These include:

• the sum of the first n terms of an arithmetic series (and it is worth memorising the special case of the sum of the first n natural numbers); 

• the sum of the first n terms of a geometric series;

• the sum to infinity of a geometric series (and I suggest you learn the special case of the sum 1+1/2+1/4+1/8+... which sums to 2; 

• and finally the binomial expansions of  (1+x)^n  and  (a+b)^n.

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You may need to use these formulae in different ways depending on the information given in the question, and be able to work forwards and backwards depending on whether you are given information about specific terms or sums or the common difference / ratio. Again you may need to hold your nerve with the algebraic manipulation required for sequences involving algebraic terms, looking for ways to simplify and/or for terms to cancel. 

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You should be familiar with sigma notation and understand how to combine sums by altering the limits of summation, and you should be able to spot arithmetic and geometric series given in sigma notation.

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Look out for questions on series which also include other functions such as trigonometric, or logarithmic, (or even algebraic expressions), as these may lead to periodic sequences. While there is no formula as such to sum these finite periodic series, you should be able to work out the sum of the block of repeating terms, and then calculate how many full blocks are needed, and if a partial block is required to make the correct number of terms.

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You should understand the notation for ’n factorial’ as well as ’n choose r’ and be able to write down their definitions. As well as more typical A level questions on binomial expansion, you may need to expand expressions such as (a+f(x))^n and you can achieve this by making a substitution such as b=f(x) and then doing an initial expansion of (a+b)^n  before substituting f(x) back into the expression and expanding further if required. However it can be useful to learn the formula for ‘trinomial’ expansion which can often give a really quick shortcut to working out the coefficients of these more complex expansions.