Arithmetic Progressions Cheat Sheet

Key Definitions

• Arithmetic Progression (AP): A sequence of numbers where the difference between consecutive terms is constant.

• Common Difference (d): The fixed difference between consecutive terms in an AP.

• n-th Term (Tn): The term at position n in the sequence.

Important Concepts

Properties of AP

• If \( a \) is the first term and \( d \) is the common difference:

• \( Tn = a + (n-1)d \)

• The sum of the first n terms (\( S_n \)):

• \( S_n = \frac{n}{2} \times (2a + (n-1)d) \) or \( S_n = \frac{n}{2} \times (a + Tn) \)

Key Formulas

• Tn: n-th term

• a: first term

• d: common difference

• n: term number

S_n = n/2 * (2a + (n-1)d)

• S_n: sum of first n terms

Common Mistakes to Avoid

Common Mistake: Forgetting to apply the formula correctly for the sum of terms; ensure you use the right \( n \).

Memory Tricks

Tip: "A Distant Star" —

• A: First term (a)

• D: Common difference (d)

• S: Sum of terms (S_n)

• T: n-th term (Tn)

Key Diagrams

   a, a+d, a+2d, a+3d, ...
   ^   ^   ^   ^   ^
   |   |   |   |   |
   1st 2nd 3rd 4th