Arithmetic Series and Natural Numbers

In the TMUA, you are expected to recognise and manipulate Arithmetic Progressions (APs). These are sequences where each term is found by adding a fixed amount, the common difference, to the previous term. This constant difference is denoted by $d$, and the very first value in the sequence is denoted by $a$.

Standard Terminology and Formulas

The fundamental definitions for an arithmetic sequence are as follows:

1. First term: $u_1 = a$
2. Common difference: $d = u_{n+1} - u_n$
3. General term ($n^{th}$ term): $u_n = a + (n - 1)d$

You should understand that $u_n$ is built by starting at $a$ and adding the difference $d$ exactly $n - 1$ times to reach the $n^{th}$ position.

Sum of an Arithmetic Series

The sum of the first $n$ terms of an arithmetic progression, denoted by $S_n$, can be expressed in two primary ways. It is highly recommended that you know how to derive these formulas, as the derivation provides insight into the structure of the series.

The first form is: $S_n = \frac{n}{2}(2a + (n - 1)d)$

The second form is often more intuitive for geometric or conceptual problems: $S_n = \frac{n}{2}(a + u_n)$

This second form reveals a powerful interpretation: the sum of an arithmetic series is equal to the number of terms multiplied by the average value of the terms. That is, $S_n = n \times \left(\frac{u_1 + u_n}{2}\right)$.

Sum of the First $n$ Natural Numbers

A specific and frequent case of an arithmetic series is the sum of the first $n$ natural numbers ($1, 2, 3, \dots, n$). In this sequence, the first term $a = 1$ and the common difference $d = 1$. The $n^{th}$ term is simply $n$.

Substituting these values into the sum formula: $S_n = \frac{n}{2}(1 + n) = \frac{n(n + 1)}{2}$

You should be prepared to use this result quickly in problems involving counting or sigma notation.

Creating New Arithmetic Series

You can generate new arithmetic series by adding or combining existing ones. If we have two arithmetic sequences: $u_n = a + (n - 1)d$ $v_n = A + (n - 1)D$

Their sum $w_n = u_n + v_n$ is also an arithmetic sequence. We can verify this by looking at the resulting expression: $u_n + v_n = (a + A) + (n - 1)(d + D)$

This new sequence has a first term of $(a + A)$ and a common difference of $(d + D)$. This property extends to any linear combination $\alpha u_n + \beta v_n$, which remains an arithmetic progression with first term $\alpha a + \beta A$ and common difference $\alpha d + \beta D$.

Counting and the Fence Post Issue

When calculating the sum of a specific portion of a sequence, be extremely careful with the number of terms. A common mistake is the fence post error. For instance, in a sum starting at the $m^{th}$ term and ending at the $n^{th}$ term, the number of terms is not $n - m$, but rather $n - m + 1$. For example, the number of terms from $u_3$ to $u_{10}$ is $10 - 3 + 1 = 8$. Always check this carefully, perhaps by counting on your fingers for small intervals, to ensure your value of $n$ in the sum formula is correct.