1.04i Geometric sequences: nth term and finite series sum

The third and fourth terms of a geometric series are 6.4 and 5.12 respectively. Find

1. the common ratio of the series, [2]
2. the first term of the series, [2]
3. the sum to infinity of the series. [2]
4. Calculate the difference between the sum to infinity of the series and the sum of the first 25 terms of the series. [4]

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]

The third and fourth terms of a geometric series are 6.4 and 5.12 respectively. Find

1. the common ratio of the series, [2]
2. the first term of the series, [2]
3. the sum to infinity of the series. [2]
4. Calculate the difference between the sum to infinity of the series and the sum of the first 25 terms of the series. [4]

A geometric series has first term \(1200\). Its sum to infinity is \(960\).

1. Show that the common ratio of the series is \(-\frac{1}{4}\). [3]
2. Find, to 3 decimal places, the difference between the ninth and tenth terms of the series. [3]
3. Write down an expression for the sum of the first \(n\) terms of the series. [2]

Given that \(n\) is odd,

1. prove that the sum of the first \(n\) terms of the series is \(960(1 + 0.25^n)\). [2]

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]

The amounts of oil pumped from an oil well in each of the years 2001 to 2004 formed a geometric progression with common ratio 0.9. The amount pumped in 2001 was 100 000 barrels.

1. Calculate the amount pumped in 2004. [2]

It is assumed that the amounts of oil pumped in future years will continue to follow the same geometric progression. Production from the well will stop at the end of the first year in which the amount pumped is less than 5000 barrels.

1. Calculate in which year the amount pumped will fall below 5000 barrels. [4]
2. Calculate the total amount of oil pumped from the well from the year 2001 up to and including the final year of production. [3]

On its first trip between Maltby and Grenlish, a steam train uses 1.5 tonnes of coal. As the train does more trips, it becomes less efficient so that each subsequent trip uses 2% more coal than the previous trip.

1. Show that the amount of coal used on the fifth trip is 1.624 tonnes, correct to 4 significant figures. [2]
2. There are 39 tonnes of coal available. An engineer wishes to calculate \(N\), the total number of trips possible. Show that \(N\) satisfies the inequality $$1.02^N < 1.52.$$ [4]
3. Hence, by using logarithms, find the greatest number of trips possible. [4]

Records are kept of the number of copies of a certain book that are sold each week. In the first week after publication 3000 copies were sold, and in the second week 2400 copies were sold. The publisher forecasts future sales by assuming that the number of copies sold each week will form a geometric progression with first two terms 3000 and 2400. Calculate the publisher's forecasts for

1. the number of copies that will be sold in the 20th week after publication, [3]
2. the total number of copies sold during the first 20 weeks after publication, [2]
3. the total number of copies that will ever be sold. [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. [1]
2. Find the sum to infinity of the series. [2]

\includegraphics{figure_12} A branching plant has stems, nodes, leaves and buds. • There are 7 leaves at each node. • From each node, 2 new stems grow. • At the end of each final stem, there is a bud. Fig. 12 shows one such plant with 3 stages of nodes. It has 15 stems, 7 nodes, 49 leaves and 8 buds.

1. One of these plants has 10 stages of nodes.
   1. How many buds does it have? [2]
   2. How many stems does it have? [2]
2. 1. Show that the number of leaves on one of these plants with \(n\) stages of nodes is $$7(2^n - 1).$$ [2]
   2. One of these plants has \(n\) stages of nodes and more than 200000 leaves. Show that \(n\) satisfies the inequality \(n > \frac{\log_{10} 200007 - \log_{10} 7}{\log_{10} 2}\). Hence find the least possible value of \(n\). [4]

A hot drink when first made has a temperature which is \(65°C\) higher than room temperature. The temperature difference, \(d °C\), between the drink and its surroundings decreases by \(1.7\%\) each minute.

1. Show that 3 minutes after the drink is made, \(d = 61.7\) to 3 significant figures. [2]
2. Write down an expression for the value of \(d\) at time \(n\) minutes after the drink is made, where \(n\) is an integer. [1]
3. Show that when \(d < 3\), \(n\) must satisfy the inequality $$n > \frac{\log_{10} 3 - \log_{10} 65}{\log_{10} 0.983}.$$ Hence find the least integer value of \(n\) for which \(d < 3\). [4]
4. The temperature difference at any time \(t\) minutes after the drink is made can also be expressed as \(d = 65 \times 10^{-kt}\), for some constant \(k\). Use the value of \(d\) for 1 minute after the drink is made to calculate the value of \(k\). Hence find the temperature difference 25.3 minutes after the drink is made. [4]

The second term of a geometric progression is 24. The sum to infinity of this progression is 150. Write down two equations in \(a\) and \(r\), where \(a\) is the first term and \(r\) is the common ratio. Solve your equations to find the possible values of \(a\) and \(r\). [5]

An arithmetic progression (AP) and a geometric progression (GP) have the same first and fourth terms as each other. The first term of both is 1.5 and the fourth term of both is 12. Calculate the difference between the tenth terms of the AP and the GP. [5]

A geometric series has common ratio \(\frac{1}{3}\). Given that the sum of the first four terms of the series is 200,

1. find the first term of the series, [3]
2. find the sum to infinity of the series. [2]

The second and fifth terms of a geometric series are \(-48\) and \(6\) respectively.

1. Find the first term and the common ratio of the series. [5]
2. Find the sum to infinity of the series. [2]
3. Show that the difference between the sum of the first \(n\) terms of the series and its sum to infinity is given by \(2^{6-n}\). [5]

A geometric series has first term 75 and second term \(-15\).

1. Find the common ratio of the series. [2]
2. Find the sum to infinity of the series. [2]

A geometric progression has third term 36 and fourth term 27. Find

1. the common ratio, [2]
2. the fifth term, [2]
3. the sum to infinity. [4]

The first three terms of a geometric series are \((x - 2)\), \((x + 6)\) and \(x^2\) respectively.

1. Show that \(x\) must be a solution of the equation $$x^3 - 3x^2 - 12x - 36 = 0. \quad \text{(I)}$$ [3]
2. Verify that \(x = 6\) is a solution of equation (I) and show that there are no other real solutions. [6]

Using \(x = 6\),

1. find the common ratio of the series, [1]
2. find the sum of the first eight terms of the series. [2]

A geometric progression has first term 75 and second term \(-15\).

1. Find the common ratio. [2]
2. Find the sum to infinity. [2]

Jill has 3 daughters and no sons. They are generation 1 of Jill's descendants. Each of her daughters has 3 daughters and no sons. Jill's 9 granddaughters are generation 2 of her descendants. Each of her granddaughters has 3 daughters and no sons; they are descendant generation 3. Jill decides to investigate what would happen if this pattern continues, with each descendant having 3 daughters and no sons.

1. How many of Jill's descendants would there be in generation 8? [2]
2. How many of Jill's descendants would there be altogether in the first 15 generations? [3]
3. After \(n\) generations, Jill would have over a million descendants altogether. Show that \(n\) satisfies the inequality $$n > \frac{\log_{10}2000003}{\log_{10}3} - 1.$$ Hence find the least possible value of \(n\). [4]
4. How many fewer descendants would Jill have altogether in 15 generations if instead of having 3 daughters, she and each subsequent descendant has 2 daughters? [3]

The second term of a geometric progression is 24. The sum to infinity of this progression is 150. Write down two equations in \(a\) and \(r\), where \(a\) is the first term and \(r\) is the common ratio. Solve your equations to find the possible values of \(a\) and \(r\). [5]

A geometric progression has first term \(a\) and common ratio \(r\). The second term is 6 and the sum to infinity is 25.

1. Write down two equations in \(a\) and \(r\). Show that one possible value of \(a\) is 10 and find the other possible value of \(a\). Write down the corresponding values of \(r\). [7]
2. Show that the ratio of the \(n\)th terms of the two geometric progressions found in part (i) can be written as \(2^{n-2} : 3^{n-2}\). [3]