Arithmetic Progressions

• Linear Relationship: The value of any term $u_n$ is a linear function of its position $n$. This means that if you were to graph the terms against their positions, they would lie on a straight line with a gradient equal to $d$.

• The $n-1$ Logic: To reach the $n^{th}$ term from the first term, you must jump forward $n-1$ times. Therefore, you add the common difference $d$ exactly $n-1$ times to the first term $a$.

• Arithmetic Mean Property: In any AP, any term (except the first) is the arithmetic mean of its immediate neighbors. For example, $u_2 = \frac{u_1 + u_3}{2}$.

Finding the General Term

• Formula: The $n^{th}$ term is calculated using > $u_n = a + (n - 1)d$
• Application: Use this when you need to find a specific term in a sequence or when you need to determine how many terms are in a finite sequence by setting $u_n$ to the last term.

Summing an Arithmetic Series

• Standard Formula: When the first term, common difference, and number of terms are known: > $S_n = \frac{n}{2}[2a + (n - 1)d]$
• Last Term Formula: When the first term ($a$) and the last term ($l$) are known: > $S_n = \frac{n}{2}(a + l)$

Solving for Unknowns

• Simultaneous Equations: If two non-consecutive terms are known (e.g., $u_5$ and $u_{12}$), set up two equations using the $u_n$ formula to solve for the variables $a$ and $d$.

• $d > 0$ vs $d < 0$: An increasing AP has a positive common difference, while a decreasing AP has a negative common difference. Always include the sign when performing calculations.

• The Method of Reversal: The formula for $S_n$ is derived by writing the sum twice: once in forward order and once in reverse order.

• Pairing Strategy: When these two sums are added vertically, every pair of terms sums to the same value: $a + l$, or $2a + (n-1)d$.

• Final Step: Since there are $n$ such pairs, the total is $n(a+l)$. Because this represents $2S_n$, we divide by 2 to reach the final formula.

• The Integer Constraint: In exam problems asking for 'which term' or 'how many terms', the value of $n$ must be a positive integer. If a calculation yields $n = 10.2$, the context usually requires rounding up or down depending on the inequality.

• Verify the Difference: Always check that $d$ is consistent between the first three terms to confirm the sequence is actually arithmetic before applying formulas.

• Formula Selection: Use $S_n = \frac{n}{2}(a+l)$ whenever possible as it is computationally simpler and less prone to arithmetic errors than the version involving $d$.

• Bracket Management: When $d$ is negative, always use brackets in the formula (e.g., $+(n-1)(-3)$) to avoid accidentally subtracting instead of multiplying.