Nội dung text 17 Arithmetic Progression.pdf
MSTC 17: ARITHMETIC PROGRESSION 1. Arithmetic Progression • Sequence – a.k.a. Progression. It is an ordered set of quantities of one-to-one correspondence with the set of positive integers. The quantities follow a specific condition. • Series – It is an expression that indicates the sum of the terms of a given sequence. A common difference distinguishes Arithmetic Progressions between adjacent terms. • Common Difference – Usually denoted as d • First Term – Usually denoted as a. 2. General Term The first few terms of an arithmetic progression are a1 = a a2 = a + d a3 = a + 2d a4 = a + 3d a5 = a + 4d Notice that from the first few terms, the coefficient of d changes, following the rule n − 1. Therefore, the general term of an arithmetic progression is an = a + (n − 1)d 3. Partial Sum Let Sn be the sum of the first n terms of the sequence. The sum Sn = a + (a + d) + (a + 2d) + ⋯ + [a + (n − 3)d] + [a + (n − 2)d] + [a + (n − 1)d] Rearrange the terms Sn = [a + (n − 1)d] + [a + (n − 2)d] + [a + (n − 3)d] + ⋯ + (a + 2d) + (a + d) + a Add the equations 2Sn = [2a + (n − 1)d] + [2a + (n − 1)d] + ⋯ + [2a + (n − 1)d] or 2Sn = n[2a + (n − 1)d] Solve for Sn Sn = n 2 [2a + (n − 1)d] Therefore, a partial sum formula is Sn = n 2 [2a + (n − 1)d] Derive the other possible formula: