Arithmetic Sequences and Series

Arithmetic Series

Arithmetic series, including the formula for the sum of the first $n$ natural numbers.

In the TMUA/ESAT we expect you to know what an arithmetic series [and sequence] is and recognise one when it appears. We often refer to these as Arithmetic Progressions [APs] in the TMUA. You should also know and understand [i.e., be able to derive] the standard formulae and terminology:

$\text{first term} = a$

$\text{common difference} = d$

$(\text{nth term}) \ u_n = a + (n - 1)d$

$(\text{sum to n terms}) \ S_n = \frac{n}{2}(2a + (n - 1)d) = \frac{n}{2}(a + a + (n - 1)d) = \frac{n}{2}(u_1 + u_n)$

The last expression for the sum can be thought of as saying:

$S_n = n \times \left(\frac{u_1 + u_n}{2}\right) = \text{number of terms} \times \text{average value of terms}$

And finally, it is useful to note that $u_{n+1} - u_n = d$.

We can create new arithmetic series by adding two [or more] series together:

$S_n = a + (a + d) + (a + 2d) + \dots + (a + (n - 1)d)$

$T_n = A + (A + D) + (A + 2D) + \dots + (A + (n - 1)D)$

And then we see that the sum is

$S_n + T_n = a + A + (a + A + d + D) + (a + A + 2(d + D)) + \dots + (a + A + (n - 1)(d + D))$

which is the sum of an arithmetic sequence with first term $a + A$ and common difference $d + D$.

(We will not test your ability to derive standard formulae in the TMUA/ESAT; but we would recommend, for your general mathematics education, that you know how to derive every formula in the TMUA/ESAT specification as the derivation will give you insights into the structure of the topic. And by "knowing how to derive" we don't mean that you have just learnt the derivation, the important point is understanding it.)

Exercise

We could also look at any linear combination of two such sequences: $\alpha u_n + \beta v_n$; verify these are also arithmetic sequences. Does this work for linear sums of more than two sequences [e.g., $\alpha u_n + \beta v_n + \gamma w_n$]?

There are lots of questions on arithmetic sequences in the TMUA/ESAT past papers. We recommend you work through the past papers first [initially under timed conditions] and if you get stuck, study the detailed worked answers we have supplied for each TMUA/ESAT paper.