1.04i Geometric sequences: nth term and finite series sum

7 The first term of a geometric progression is 81 and the fourth term is 24 . Find

1. the common ratio of the progression,
2. the sum to infinity of the progression. The second and third terms of this geometric progression are the first and fourth terms respectively of an arithmetic progression.
3. Find the sum of the first ten terms of the arithmetic progression.

7

1. Find the sum to infinity of the geometric progression with first three terms \(0.5,0.5 ^ { 3 }\) and \(0.5 ^ { 5 }\).
2. The first two terms in an arithmetic progression are 5 and 9. The last term in the progression is the only term which is greater than 200 . Find the sum of all the terms in the progression.

1 The first term of a geometric progression is 12 and the second term is - 6 . Find

1. the tenth term of the progression,
2. the sum to infinity.

7

1. The first two terms of an arithmetic progression are 1 and \(\cos ^ { 2 } x\) respectively. Show that the sum of the first ten terms can be expressed in the form \(a - b \sin ^ { 2 } x\), where \(a\) and \(b\) are constants to be found.
2. The first two terms of a geometric progression are 1 and \(\frac { 1 } { 3 } \tan ^ { 2 } \theta\) respectively, where \(0 < \theta < \frac { 1 } { 2 } \pi\).
   1. Find the set of values of \(\theta\) for which the progression is convergent.
   2. Find the exact value of the sum to infinity when \(\theta = \frac { 1 } { 6 } \pi\).

4 The third term of a geometric progression is - 108 and the sixth term is 32 . Find

1. the common ratio,
2. the first term,
3. the sum to infinity.

10

1. The first and last terms of an arithmetic progression are 12 and 48 respectively. The sum of the first four terms is 57. Find the number of terms in the progression.
2. The third term of a geometric progression is four times the first term. The sum of the first six terms is \(k\) times the first term. Find the possible values of \(k\).

9

1. In an arithmetic progression, the sum, \(S _ { n }\), of the first \(n\) terms is given by \(S _ { n } = 2 n ^ { 2 } + 8 n\). Find the first term and the common difference of the progression.
2. The first 2 terms of a geometric progression are 64 and 48 respectively. The first 3 terms of the geometric progression are also the 1st term, the 9th term and the \(n\)th term respectively of an arithmetic progression. Find the value of \(n\).

6 The 1st, 2nd and 3rd terms of a geometric progression are the 1st, 9th and 21st terms respectively of an arithmetic progression. The 1st term of each progression is 8 and the common ratio of the geometric progression is \(r\), where \(r \neq 1\). Find

1. the value of \(r\),
2. the 4th term of each progression.

2 The first term in a progression is 36 and the second term is 32 .

1. Given that the progression is geometric, find the sum to infinity.
2. Given instead that the progression is arithmetic, find the number of terms in the progression if the sum of all the terms is 0 .

9

1. The first term of a geometric progression in which all the terms are positive is 50 . The third term is 32 . Find the sum to infinity of the progression.
2. The first three terms of an arithmetic progression are \(2 \sin x , 3 \cos x\) and ( \(\sin x + 2 \cos x\) ) respectively, where \(x\) is an acute angle.
   1. Show that \(\tan x = \frac { 4 } { 3 }\).
   2. Find the sum of the first twenty terms of the progression.

9 A water tank holds 2000 litres when full. A small hole in the base is gradually getting bigger so that each day a greater amount of water is lost.

1. On the first day after filling, 10 litres of water are lost and this increases by 2 litres each day.
   1. How many litres will be lost on the 30th day after filling?
   2. The tank becomes empty during the \(n\)th day after filling. Find the value of \(n\).
   3. Assume instead that 10 litres of water are lost on the first day and that the amount of water lost increases by \(10 \%\) on each succeeding day. Find what percentage of the original 2000 litres is left in the tank at the end of the 30th day after filling.
      [0pt] [Questions 10 and 11 are printed on the next page.]

4 The 1st, 3rd and 13th terms of an arithmetic progression are also the 1st, 2nd and 3rd terms respectively of a geometric progression. The first term of each progression is 3 . Find the common difference of the arithmetic progression and the common ratio of the geometric progression.

4

1. An arithmetic progression has a first term of 32, a 5th term of 22 and a last term of - 28 . Find the sum of all the terms in the progression.
2. Each year a school allocates a sum of money for the library. The amount allocated each year increases by \(2.5 \%\) of the amount allocated the previous year. In 2005 the school allocated \(\\) 2000$. Find the total amount allocated in the years 2005 to 2014 inclusive.

7

1. The first two terms of an arithmetic progression are 16 and 24. Find the least number of terms of the progression which must be taken for their sum to exceed 20000.
2. A geometric progression has a first term of 6 and a sum to infinity of 18. A new geometric progression is formed by squaring each of the terms of the original progression. Find the sum to infinity of the new progression.

8

1. A geometric progression has a second term of 12 and a sum to infinity of 54 . Find the possible values of the first term of the progression.
2. The \(n\)th term of a progression is \(p + q n\), where \(p\) and \(q\) are constants, and \(S _ { n }\) is the sum of the first \(n\) terms.
   1. Find an expression, in terms of \(p , q\) and \(n\), for \(S _ { n }\).
   2. Given that \(S _ { 4 } = 40\) and \(S _ { 6 } = 72\), find the values of \(p\) and \(q\).

3 A company producing salt from sea water changed to a new process. The amount of salt obtained each week increased by \(2 \%\) of the amount obtained in the preceding week. It is given that in the first week after the change the company obtained 8000 kg of salt.

1. Find the amount of salt obtained in the 12th week after the change.
2. Find the total amount of salt obtained in the first 12 weeks after the change.

3 The common ratio of a geometric progression is 0.99 . Express the sum of the first 100 terms as a percentage of the sum to infinity, giving your answer correct to 2 significant figures.

8

1. The third and fourth terms of a geometric progression are 48 and 32 respectively. Find the sum to infinity of the progression.
2. Two schemes are proposed for increasing the amount of household waste that is recycled each week. Scheme \(A\) is to increase the amount of waste recycled each month by 0.16 tonnes.
   Scheme \(B\) is to increase the amount of waste recycled each month by \(6 \%\) of the amount recycled in the previous month.
   The proposal is to operate the scheme for a period of 24 months. The amount recycled in the first month is 2.5 tonnes. For each scheme, find the total amount of waste that would be recycled over the 24 -month period. Scheme \(A\) Scheme \(B\) \(\_\_\_\_\)

10

1. In an arithmetic progression, the sum of the first ten terms is equal to the sum of the next five terms. The first term is \(a\).
   1. Show that the common difference of the progression is \(\frac { 1 } { 3 } a\).
   2. Given that the tenth term is 36 more than the fourth term, find the value of \(a\).
2. The sum to infinity of a geometric progression is 9 times the sum of the first four terms. Given that the first term is 12 , find the value of the fifth term.

7 The function f is defined for \(x \geqslant 0\) by \(\mathrm { f } ( x ) = ( 4 x + 1 ) ^ { \frac { 3 } { 2 } }\).

1. Find \(\mathrm { f } ^ { \prime } ( x )\) and \(\mathrm { f } ^ { \prime \prime } ( x )\).
   The first, second and third terms of a geometric progression are respectively \(\mathrm { f } ( 2 ) , \mathrm { f } ^ { \prime } ( 2 )\) and \(k \mathrm { f } ^ { \prime \prime } ( 2 )\).
2. Find the value of the constant \(k\).

2 A geometric progression, for which the common ratio is positive, has a second term of 18 and a fourth term of 8 . Find

1. the first term and the common ratio of the progression,
2. the sum to infinity of the progression.

2 Find

1. the sum of the first ten terms of the geometric progression \(81,54,36 , \ldots\),
2. the sum of all the terms in the arithmetic progression \(180,175,170 , \ldots , 25\).

6 A small trading company made a profit of \(\\) 250000\( in the year 2000. The company considered two different plans, plan \)A\( and plan \)B$, for increasing its profits. Under plan \(A\), the annual profit would increase each year by \(5 \%\) of its value in the preceding year. Find, for plan \(A\),

1. the profit for the year 2008,
2. the total profit for the 10 years 2000 to 2009 inclusive. Under plan \(B\), the annual profit would increase each year by a constant amount \(\\) D\(.
3. Find the value of \)D$ for which the total profit for the 10 years 2000 to 2009 inclusive would be the same for both plans.

6

1. Find the sum of all the integers between 100 and 400 that are divisible by 7 .
2. The first three terms in a geometric progression are \(144 , x\) and 64 respectively, where \(x\) is positive. Find
   1. the value of \(x\),
   2. the sum to infinity of the progression.

4 The 1st term of an arithmetic progression is \(a\) and the common difference is \(d\), where \(d \neq 0\).

1. Write down expressions, in terms of \(a\) and \(d\), for the 5th term and the 15th term. The 1st term, the 5th term and the 15th term of the arithmetic progression are the first three terms of a geometric progression.
2. Show that \(3 a = 8 d\).
3. Find the common ratio of the geometric progression.