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Problems in Arithmetic and Geometrical Series

Three numbers are in arithmetic progression. The sum of the three numbers is 30 and the sum of their squares is 398. What are the three numbers?

b) An arithmetic series is such that its first term is a and its third term is b. The sum of the first n terms is Sn . Find S4 in terms of a and b. Given that S4, S5 , S7 are consecutive terms of a geometric series, show that 7a^2=13b^2 .

c) Given that the roots of the equation 3x^3 - 7x + px + 24=0 are in geometric progression, find the value of p and solve the equation.