Arithmetic Progression

Chapter in a nutshell: An Arithmetic Progression (AP) is a sequence where each term differs from the previous by a fixed common difference $d$. You must find the $n$th term and the sum of $n$ terms, and use clever ways of choosing terms in problems.

1. Key Ideas

• AP: $a,\ a+d,\ a+2d,\ \dots$ — constant common difference $d = a_2-a_1$.
• First term $a$ (or $a_1$); $n$th (general) term $a_n$; last term $l$; number of terms $n$.

2. Formulas (quick reference — all of them)

• $n$th term: $\;a_n = a+(n-1)d$
• Common difference: $\;d=a_n-a_{n-1}$
• Last term: $\;l=a+(n-1)d$
• Number of terms: $\;n=\dfrac{l-a}{d}+1$
• Sum of $n$ terms: $\;S_n=\dfrac{n}{2}\big[\,2a+(n-1)d\,\big]$
• Sum (using last term): $\;S_n=\dfrac{n}{2}\,(a+l)$
• $n$th term from the sum: $\;a_n=S_n-S_{n-1}$
• Sum of first $n$ natural numbers: $\;1+2+\dots+n=\dfrac{n(n+1)}{2}$
• Sum of first $n$ odd numbers $=n^2$; of first $n$ even numbers $=n(n+1)$.
• Arithmetic mean of a, b: $\dfrac{a+b}{2}$.

3. Choosing Terms Cleverly (technique)

• 3 terms in AP: $a-d,\ a,\ a+d$ (sum = 3a).
• 4 terms in AP: $a-3d,\ a-d,\ a+d,\ a+3d$ (common difference 2d).
• 5 terms: $a-2d,\ a-d,\ a,\ a+d,\ a+2d$.

4. Prerequisite Ideas

• Meaning of a sequence/series; basic linear equations (to solve for a, d, n).

5. Worked Example (one, for the method)

6. Common Mistakes to Avoid

• Using $n$ instead of $(n-1)$ in the $n$th-term formula.
• Forgetting $d$ can be negative (decreasing AP).
• Mixing $S_n=\frac{n}{2}[2a+(n-1)d]$ with $S_n=\frac{n}{2}(a+l)$ — use $l$ only when the last term is known.
• Wrong count of terms: $n=\frac{l-a}{d}+1$ (don't forget the +1).

7. Likely Exam Questions (with crisp answers)

1. What is an AP? → A sequence with a constant common difference.
2. Write the formula for the $n$th term. → $a_n=a+(n-1)d$.
3. Write both sum formulas. → $S_n=\frac{n}{2}[2a+(n-1)d]=\frac{n}{2}(a+l)$.
4. How do you find $d$? → Subtract any term from the next ($a_n-a_{n-1}$).
5. How are 3 numbers in AP usually taken? → $a-d,\ a,\ a+d$.
6. Find $a_n$ from sums. → $a_n=S_n-S_{n-1}$.
7. Sum of first $n$ natural numbers? → $\frac{n(n+1)}{2}$.
8. AM of two numbers a and b? → $\frac{a+b}{2}$.
9. Number of terms from a to l with difference d? → $\frac{l-a}{d}+1$.
10. Sum of first $n$ odd numbers? → $n^2$.

Extended & Prerequisite Formulas (full)

• $a_n=a+(n-1)d$; $S_n=\tfrac{n}{2}[2a+(n-1)d]=\tfrac{n}{2}(a+l)$; $a_n=S_n-S_{n-1}$; $n=\tfrac{l-a}{d}+1$.
• 3 terms: a−d,a,a+d; 4 terms: a−3d,a−d,a+d,a+3d.
• Standard sums (prereq): $\sum n=\tfrac{n(n+1)}{2}$; $\sum n^2=\tfrac{n(n+1)(2n+1)}{6}$; $\sum n^3=\left[\tfrac{n(n+1)}{2}\right]^2$; first n odd = n²; first n even = n(n+1).
• Arithmetic mean of a,b = (a+b)/2.