An arithmetic progression is a sequence of numbers in which the difference between a term and its preceding term is always constant. This constant difference is called the common difference.For example, consider the sequence: $2, 5, 8, 11, 14, 17, \dots$$a_1 = 2,$$a_2 = 5,$$a_3 = 8,$$a_4 = 11$Now check the differences:$a2−a1=5−2=3$;$a3−a2=8−5=3$;$a4−a3=11−8=3$Result: Since differences are equal ($3$), it is an A.P.
General Form & $n^{th}$ Term in AP
Standard Notation: $a, a+d, a+2d, a+3d, \dots$$a$ = first term$d$ = common differenceTerm Formulas:$1^{st}$ term: $a_1 = a$$2^{nd}$ term: $a_2 = a+d$$3^{rd}$ term: $a_3 = a+2d$General Formula ($n^{th}$ term):$$a_n = a + (n-1)d$$(Applies to any variable: $a_p = a + (p-1)d$ or $a_q = a + (q-1)d$)
3. Sum of Terms ($S_n$)
Formula for sum of first $n$ terms:$$S_n = \frac{n}{2}[2a + (n-1)d]$$Finding $a_n$ from $S_n$: If the sum is known, the $n^{th}$ term is:$$a_n = S_n – S_{n-1}$$
4. Properties of Three Terms ($a, b, c$)
If $a, b, c$ are in A.P., then common differences must be equal:$b – a = c – b$Simplified Relationship:$$2b = a + c$$Key takeaway: The middle term ($b$) is the average of the first ($a$) and third ($c$) terms.