In an arithmetic progression, the sum of the first \(30\) terms is \(-1065\). The sum of the next \(20\) terms is \(-2210\). Find the first term and the common difference.[5]
A geometric progression is such that the first term is \(4\) and the sum of the first three terms is \(7\). Find the two possible values of the common ratio and find the sum to infinity for the convergent progression.[5]
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Answer:(a) First term \(8\), common difference \(-3\). (b) \(r=\dfrac12,-\dfrac32\); convergent sum \(8\).