An arithmetic progression has first term \(A\) and common difference \(D\). The sum of its first two terms is 25 and the sum of its first four terms is 250 .
(A) Find the values of \(A\) and \(D\).
(B) Find the sum of the 21st to 50th terms inclusive of this sequence.
A geometric progression has first term \(a\) and common ratio \(r\), with \(r \neq \pm 1\). The sum of its first two terms is 25 and the sum of its first four terms is 250 .
Use the formula for the sum of a geometric progression to show that \(\frac { r ^ { 4 } - 1 } { r ^ { 2 } - 1 } = 10\) and hence or otherwise find algebraically the possible values of \(r\) and the corresponding values of \(a\).