Find specific nth term

6 The second term of a geometric progression is 54 and the sum to infinity of the progression is 243 . The common ratio is greater than \(\frac { 1 } { 2 }\). Find the tenth term, giving your answer in exact form.

4 A geometric progression is such that the third term is 1764 and the sum of the second and third terms is 3444 . Find the 50th term.

4 Sequences \(\mathrm { A } , \mathrm { B }\) and C are shown below. They each continue in the pattern established by the given terms. $$\begin{array} { l l l l l l l l l } \text { A: } & 1 , & 2 , & 4 , & 16 , & 32 , & \ldots & \\ \text { B: } & 20 , & - 10 , & 5 , & - 2.5 , & 1.25 , & - 0.625 , & \ldots \\ \text { C: } & 20 , & 5 , & 1 , & 20 , & 5 , & 1 , & \ldots \end{array}$$

1. Which of these sequences is periodic?
2. Which of these sequences is convergent?
3. Find, in terms of \(n\), the \(n\)th term of sequence A .

2 The first term of a geometric series is 5.4 and the common ratio is 0.1.

1. Find the fourth term of the series.
2. Find the sum to infinity of the series.

2 The first three terms of a geometric progression are 4, 2, 1.
Find the twentieth term, expressing your answer as a power of 2 .
Find also the sum to infinity of this progression.

5 The second term of a geometric sequence is 6 and the fifth term is - 48 .
Find the tenth term of the sequence.
Find also, in simplified form, an expression for the sum of the first \(n\) terms of this sequence.

3 The first term of a geometric progression is 50 and the common ratio is 0.9 .

1. Find the fifth term.
2. Find the sum of the first thirty terms.
3. Find the sum to infinity.

The second term of a geometric sequence is 6 and the fifth term is \(-48\). Find the tenth term of the sequence. Find also, in simplified form, an expression for the sum of the first \(n\) terms of this sequence. [5]