Arithmetic Sequence Formulae and Subscript Notation

The Causal Logic of the Formula

To understand how the general formula is built, we can look at how the common difference is added to reach each position ($n$):

• To find the first term ($u_1$), you start with $a$. The common difference has been added zero times.
• To find the second term ($u_2$), you take $a$ and add $d$ exactly once: $a + d$.
• To find the third term ($u_3$), you take $a$ and add $d$ twice: $a + 2d$.
• Notice the pattern: the number of times you add $d$ is always exactly one less than the position number ($n$).
• Therefore, to find any term, you must add the common difference $(n - 1)$ times to the first term.

$$u_n = a + (n - 1)d$$

Using Subscript Notation

• Just as house numbers tell you exactly where a building sits on a street, small numbers attached to letters tell you exactly where a value sits in a pattern.
• In Higher Tier papers, you must use subscript notation.
• The index (the small letter or number, like $n$) indicates the specific placement of a term within the sequence. $n$ must always be a positive integer.
• $u_1$ means the first term, $u_n$ means the current term at position , and means the very next term in the sequence.

Worked Example: Position-to-Term Rule

What is the $15^{th}$ term of the sequence defined by the position-to-term rule $u_n = 6n - 2$?

Step 1: Identify the position index.

• $n = 15$

Step 2: Substitute $n$ into the position-to-term formula.

• $u_{15} = 6(15) - 2$

Step 3: Calculate the final value.

• $u_{15} = 90 - 2 = 88$

Term-to-Term Rules and Recurrence Relations

• Some patterns are like a chain reaction, where the next event depends entirely on the one that just happened.
• A term-to-term rule defines how to calculate the next term ($u_{n+1}$) using only the value of the previous term ($u_n$).
• This type of rule is also known as a recurrence relation or an iterative formula.
• For an arithmetic sequence, the recurrence relation is written simply as $$, but OCR questions often involve more complex linear relations.

Worked Example: Recurrence Relation

A sequence is defined by the recurrence relation $u_{n+1} = 3u_n - 4$. Given that the first term $u_1 = 5$, calculate and .

Step 1: Use $n=1$ to find $u_2$ by substituting the value of $u_1$.

• $u_2 = 3(5) - 4$

Step 2: Calculate the value of $u_2$.

• $u_2 = 15 - 4 = 11$

Step 3: Use $n=2$ to find $u_3$ by substituting your new value for $u_2$.

• $u_3 = 3(11) - 4$

Step 4: Calculate the value of $u_3$.

• $u_3 = 33 - 4 = 29$