Two related arithmetic progressions

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1. A geometric progression is such that the second term is equal to \(24 \%\) of the sum to infinity. Find the possible values of the common ratio.
2. An arithmetic progression \(P\) has first term \(a\) and common difference \(d\). An arithmetic progression \(Q\) has first term 2( \(a + 1\) ) and common difference ( \(d + 1\) ). It is given that $$\frac { 5 \text { th term of } P } { 12 \text { th term of } Q } = \frac { 1 } { 3 } \quad \text { and } \quad \frac { \text { Sum of first } 5 \text { terms of } P } { \text { Sum of first } 5 \text { terms of } Q } = \frac { 2 } { 3 } .$$ Find the value of \(a\) and the value of \(d\).

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1. 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 21 st to 50 th terms inclusive of this sequence.
2. 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 \(a\).

2．The sum of the first \(p\) terms of an arithmetic series is \(q\) and the sum of the first \(q\) terms of the same arithmetic series is \(p\) ，where \(p\) and \(q\) are positive integers and \(p \neq q\) ． Giving simplified answers in terms of \(p\) and \(q\) ，find
（a）the common difference of the terms in this series，
（b）the first term of the series，
（c）the sum of the first \(( p + q )\) terms of the series．

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1. The ninth term of an arithmetic series is 3 The sum of the first \(n\) terms of the series is \(S _ { n }\) and \(S _ { 21 } = 42\) Find the first term and common difference of the series.
   [0pt] [4 marks]
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2. A second arithmetic series has first term - 18 and common difference \(\frac { 3 } { 4 }\) The sum of the first \(n\) terms of this series is \(T _ { n }\) Find the value of \(n\) such that \(T _ { n } = S _ { n }\) [0pt] [3 marks]