(a) Each person has two parents, four grandparents, eight great-grandparents, and so on. By summing a geometric sequence, find the total number of ancestors a person has going back (1) five generations, (2) 10 generations. normal growth pattern for children aged 3-11 follows an arithmetic sequence. Given a child measures 98.2 cm at age 3, and 109.8 cm at age 5, what is the common difference of the arithmetic sequence? What would the child's height at age 8 be?

Answer:The person has62 ancestors going back five generations.The person has 2046 ancestors going back ten generations.---------------------The child's height at age 8 would be 127.2 cm.Step-by-step explanation:The first sequence is a geometric sequence.In a geometric sequence, each term is found by multiplying the previous term by a constant r.We write a geometric sequence like this:[tex]{a, ar, ar^{2}, ar^{3},...}[/tex]Where a is the first term and r is the commom factor.The sum of the first n elements of a geometric sequence is:[tex]S = \frac{a(1 - r^{n})}{1-r}[/tex]So, for the first exercise, our geometric sequence is:{2,4,8,...},so a = 2 and r = 2.1)Find the total number of ancestors a person has going back five generationsS when n = 5, so:[tex]S = \frac{a(1 - r^{n})}{1-r} = \frac{2*(1-2^{5})}{1-2} = 62[/tex]The person has 62 ancestors going back five generations.2) Going back 10 generations:S when n = 10, so:[tex]S = \frac{a(1 - r^{n})}{1-r} = \frac{2*(1-2^{10})}{1-2} = 2046[/tex]The person has 2046 ancestors going back ten generations.-------------The following question is related to an arithmetic sequence:An arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant.If the first term of an arithmetic sequence is a1 and the common difference is d, then the nth term of the sequence is given by:[tex]a_{n} = a_{1} + (n-1)d[/tex].We have the following sequence[tex]98.2, a_{2}, 109.8, a_{4}, a_{5}, a_{6}[/tex], in which [tex]a_{6}[/tex] is the child's height at age 8.We have that:[tex]d = \frac{109.8 - 98.2}{2} = 5.8[/tex]So[tex]a_{6} = a_{1} + 5d = 98.2 + 5*5.8 = 127.2[/tex].The child's height at age 8 would be 127.2 cm.