Arithmetic and Geometric Progressions APGP （算术和几何进展APGP）.pdfVIP

Arithmetic and Geometric Progressions (AP/GP) Summary 1. Arithmetic Progression (AP) : • characterized by a common difference d nth term T = a+ (n−1) d , where a is the first term of the series n n Sum to nterms S = (a+ l) , where ldenotes the last term of the series of nterms n 2 or S = n[2a+ (n−1) d] (to be used if value of lis not readily available) n 2 2. Geometric Progression (GP) : • characterized by a common ratio r n−1 nth term T = ar , where a is the first term of the series n n n a(r −1) a(1− r ) Sum to nterms S = = n r−1 1− r Note: If ris negative, then the sequence comprises terms alternating between positive and negative ones. Also, if | r |1, then the series is a convergent one, ie T → k ( kis a real n a value) as n → ∞, and it possesses a sum to infinity given by S = . ∞ 1− r 3. Techniques relevant to problem solving: (i) Forming an equation relating three terms of a sequence. If x, y and z are consecutive terms of an AP, then we have y − x = z − y z y If x, y and z are consecutive terms of a GP, then we have =