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0ATA adjusted death rate age distribution age groups age interval age-specific death rates approximation Assume assumption average birth rate calculate census Chapter characteristic equation constant corresponding matrix crude death rate difference differential equation digraph direct method dt J0 e-rxl(x equal error expj expression female fertility fi(x force of mortality formula fraction functional iteration geometric geometric progression given graph growth Hence hyperbola initial integral intrinsic rate irreducible iteration Keyfitz lived logarithm logistic curve LPRINT Makeham's male Markov chain Mathematics matrix of Problem mean age Metropolitan Statistical Area Muncie number of births number of children obtain percent population aged primitive Problem 17 Problem 22 projection matrix Prove quadratic random variable rate of increase REA0 rural Society of Actuaries solution stable population standard population subpopulations survivorship t)dt ultimate ratio urban values variance vector zero