Arithmetic Sequences and SeriesLesson · BeginnerAlgebra: Sequences & Seriesan=a1+(n−1)d,n≥1a_n=a_1+(n-1)d,\qquad n\ge 1an=a1+(n−1)d,n≥1Sn=a1+a2+⋯+an=n2(a1+an)S_n=a_1+a_2+\cdots+a_n=\frac{n}{2}(a_1+a_n)Sn=a1+a2+⋯+an=2n(a1+an)an=an−1+an+12a_n=\frac{a_{n-1}+a_{n+1}}{2}an=2an−1+an+1Number of elements=last−firstd+1\text{Number of elements}=\frac{\text{last}-\text{first}}{d}+1Number of elements=dlast−first+1Prerequisites (2)Sums of PowersGauss's Pairing Trick (Rainbow Idea)ProofsSn=a1+a2+⋯+an=n2(a1+an)S_n=a_1+a_2+\cdots+a_n=\frac{n}{2}(a_1+a_n)Sn=a1+a2+⋯+an=2n(a1+an)Sn=a1+a2+⋯+an−1+anSn=an+an−1+⋯+a2+a1 \begin{aligned} S_n&=a_1+a_2+\cdots+a_{n-1}+a_n\\ S_n&=a_n+a_{n-1}+\cdots+a_2+a_1 \end{aligned} SnSn=a1+a2+⋯+an−1+an=an+an−1+⋯+a2+a12Sn=(a1+an)+(a2+an−1)+⋯+(an−1+a2)+(an+a1) 2S_n=(a_1+a_n)+(a_2+a_{n-1})+\cdots+(a_{n-1}+a_2)+(a_n+a_1) 2Sn=(a1+an)+(a2+an−1)+⋯+(an−1+a2)+(an+a1)a2+an−1=a1+d+an−d=a1+ana_2+a_{n-1}=a_1+d+a_n-d=a_1+a_na2+an−1=a1+d+an−d=a1+ana3+an−2=a1+2d+an−2d=a1+ana_3+a_{n-2}=a_1+2d+a_n-2d=a_1+a_na3+an−2=a1+2d+an−2d=a1+anak+1+an−k=a1+kd+an−kd=a1+ana_{k+1}+a_{n-k}=a_1+kd+a_n-kd=a_1+a_nak+1+an−k=a1+kd+an−kd=a1+an2Sn=n(a1+an)⇒Sn=n(a1+an)22S_n=n(a_1+a_n)\Rightarrow S_n=\frac{n(a_1+a_n)}{2}2Sn=n(a1+an)⇒Sn=2n(a1+an)an=an−1+an+12a_n=\frac{a_{n-1}+a_{n+1}}{2}an=2an−1+an+1an−1+an+12=(an−d)+(an+d)2=2an2=an \frac{a_{n-1}+a_{n+1}}{2} = \frac{(a_n-d)+(a_n+d)}{2} = \frac{2a_n}{2} = a_n 2an−1+an+1=2(an−d)+(an+d)=22an=anNumber of elements=last−firstd+1\text{Number of elements}=\frac{\text{last}-\text{first}}{d}+1Number of elements=dlast−first+1a1,a2,…,ana_1,a_2,\ldots,a_na1,a2,…,anlast−firstd+1=an−a1d+1=a1+(n−1)d−a1d+1 \frac{\text{last}-\text{first}}{d}+1 = \frac{a_n-a_1}{d}+1 = \frac{a_1+(n-1)d-a_1}{d}+1 dlast−first+1=dan−a1+1=da1+(n−1)d−a1+1=(n−1)dd+1=n−1+1=n =\frac{(n-1)d}{d}+1=n-1+1=n =d(n−1)d+1=n−1+1=nExamplesExample 1Calculate7+10+13+⋯+527+10+13+\cdots+527+10+13+⋯+52Solutiond=10−7=3d=10-7=3d=10−7=3n=last−firstd+1=52−73+1=16 n=\frac{\text{last}-\text{first}}{d}+1 =\frac{52-7}{3}+1 =16 n=dlast−first+1=352−7+1=16Sn=n2(a1+an)=162(7+52)=472 S_n=\frac{n}{2}(a_1+a_n) =\frac{16}{2}(7+52) =472 Sn=2n(a1+an)=216(7+52)=472Related Problems (6)AMC 10B 2022 (Problem 15)MATHCOUNTS 2026 State Target Round (Problem 5)AMC 10A 2025 (Problem 11)AMC 10B/12B 2025 (Problem 17/14)AMC 8 2026 (Problem 14)AMC 10A 2012 (Problem 22)