Prove sum formula

14. A geometric series has a first term \(a\) and a common ratio \(r\).

1. Prove that the sum of the first \(n\) terms of this series is given by $$S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }$$ A liquid is to be stored in a barrel. Due to evaporation, the volume of the liquid in a barrel at the end of a year is \(7 \%\) less than the volume at the start of the year. At the start of the first year, a barrel is filled with 180 litres of the liquid.
2. Show that the amount of the liquid in this barrel at the end of 5 years is approximately 125.2 litres. At the start of each year a new identical barrel is filled with 180 litres of the liquid so that, at the end of 20 years, there are 20 barrels containing varying amounts of the liquid.
3. Calculate the total amount of the liquid, to the nearest litre, in the 20 barrels at the end of 20 years.

8. A geometric series has first term \(a\) and common ratio \(r\).

1. Prove that the sum of the first \(n\) terms of this series is given by $$S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }$$ The second term of a geometric series is - 320 and the fifth term is \(\frac { 512 } { 25 }\)
2. Find the value of the common ratio.
3. Hence find the sum of the first 13 terms of the series, giving your answer to 2 decimal places.

9. (a) A geometric series has first term \(a\) and common ratio \(r\). Prove that the sum of the first \(n\) terms of the series is $$\frac { a \left( 1 - r ^ { n } \right) } { 1 - r } .$$ Mr. King will be paid a salary of \(\pounds 35000\) in the year 2005 . Mr. King's contract promises a \(4 \%\) increase in salary every year, the first increase being given in 2006, so that his annual salaries form a geometric sequence.
(b) Find, to the nearest \(\pounds 100\), Mr. King's salary in the year 2008. Mr. King will receive a salary each year from 2005 until he retires at the end of 2024.
(c) Find, to the nearest \(\pounds 1000\), the total amount of salary he will receive in the period from 2005 until he retires at the end of 2024.

1. In this question you must show all stages of your working.

\section*{Solutions relying entirely on calculator technology are not acceptable.} A geometric series has common ratio \(r\) and first term \(a\).
Given \(r \neq 1\) and \(a \neq 0\)

1. prove that $$S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }$$ Given also that \(S _ { 10 }\) is four times \(S _ { 5 }\)
2. find the exact value of \(r\).

7. A geometric series is \(a + a r + a r ^ { 2 } + \ldots\)

1. Prove that the sum of the first \(n\) terms of this series is given by \(S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }\). The second and fourth terms of the series are 3 and 1.08 respectively.
   Given that all terms in the series are positive, find
2. the value of \(r\) and the value of \(a\),
3. the sum to infinity of the series. \includegraphics[max width=\textwidth, alt={}, center]{1033051d-18bf-4734-a556-4c8e1c789992-4_764_1159_294_299} Fig. 2 shows part of the curve with equation \(y = x ^ { 3 } - 6 x ^ { 2 } + 9 x\). The curve touches the \(x\)-axis at \(A\) and has a maximum turning point at \(B\).
   1. Show that the equation of the curve may be written as \(y = x ( x - 3 ) ^ { 2 }\), and hence write down the coordinates of \(A\).
   2. Find the coordinates of \(B\). The shaded region \(R\) is bounded by the curve and the \(x\)-axis.
   3. Find the area of \(R\).

A geometric series is \(a + ar + ar^2 + \ldots\)

1. Prove that the sum of the first \(n\) terms of this series is given by $$S_n = \frac{a(1 - r^n)}{1 - r}.$$ [4]

The second and fourth terms of the series are 3 and 1.08 respectively. Given that all terms in the series are positive, find

1. the value of \(r\) and the value of \(a\), [5]
2. the sum to infinity of the series. [3]

A geometric series is \(a + ar + ar^2 + \ldots\)

1. Prove that the sum of the first \(n\) terms of this series is given by $$S_n = \frac{a(1-r^n)}{1-r}$$ [4]

The second and fourth terms of the series are 3 and 1.08 respectively. Given that all terms in the series are positive, find

1. the value of \(r\) and the value of \(a\), [5]
2. the sum to infinity of the series. [3]

A geometric series is \(a + ar + ar^2 + \ldots\)

1. Prove that the sum of the first \(n\) terms of this series is \(S_n = \frac{a(1 - r^n)}{1 - r}\). [4]

The first and second terms of a geometric series \(G\) are 10 and 9 respectively.

1. Find, to 3 significant figures, the sum of the first twenty terms of \(G\). [3]
2. Find the sum to infinity of \(G\). [2]

Another geometric series has its first term equal to its common ratio. The sum to infinity of this series is 10.

1. Find the exact value of the common ratio of this series. [3]

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. A geometric series has common ratio \(r\) and first term \(a\). Given \(r \neq 1\) and \(a \neq 0\)

1. prove that $$S_n = \frac{a(1-r^n)}{1-r}$$ [4]

Given also that \(S_{10}\) is four times \(S_5\)

1. find the exact value of \(r\). [4]