1.04j Sum to infinity: convergent geometric series |r|<1

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.

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 an arithmetic progression is - 2222 and the common difference is 17 . Find the value of the first positive term.
2. The first term of a geometric progression is \(\sqrt { } 3\) and the second term is \(2 \cos \theta\), where \(0 < \theta < \pi\). Find the set of values of \(\theta\) for which the progression is convergent.

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.

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 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.

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.

3

1. A debt of \(\\) 3726\( is repaid by weekly payments which are in arithmetic progression. The first payment is \)\\( 60\) and the debt is fully repaid after 48 weeks. Find the third payment.
2. Find the sum to infinity of the geometric progression whose first term is 6 and whose second term is 4 .

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.

7 The equation of a curve is \(y = \frac { 12 } { x ^ { 2 } + 3 }\).

1. Obtain an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find the equation of the normal to the curve at the point \(P ( 1,3 )\).
3. A point is moving along the curve in such a way that the \(x\)-coordinate is increasing at a constant rate of 0.012 units per second. Find the rate of change of the \(y\)-coordinate as the point passes through \(P\).

5

1. The first and second terms of an arithmetic progression are 161 and 154 respectively. The sum of the first \(m\) terms is zero. Find the value of \(m\).
2. A geometric progression, in which all the terms are positive, has common ratio \(r\). The sum of the first \(n\) terms is less than \(90 \%\) of the sum to infinity. Show that \(r ^ { n } > 0.1\).

9

1. A geometric progression has first term 100 and sum to infinity 2000. Find the second term.
2. An arithmetic progression has third term 90 and fifth term 80 .
   1. Find the first term and the common difference.
   2. Find the value of \(m\) given that the sum of the first \(m\) terms is equal to the sum of the first ( \(m + 1\) ) terms.
   3. Find the value of \(n\) given that the sum of the first \(n\) terms is zero.

6

1. The sixth term of an arithmetic progression is 23 and the sum of the first ten terms is 200 . Find the seventh term.
2. A geometric progression has first term 1 and common ratio \(r\). A second geometric progression has first term 4 and common ratio \(\frac { 1 } { 4 } r\). The two progressions have the same sum to infinity, \(S\). Find the values of \(r\) and \(S\).

8

1. In a geometric progression, all the terms are positive, the second term is 24 and the fourth term is \(13 \frac { 1 } { 2 }\). Find
   1. the first term,
   2. the sum to infinity of the progression.
2. A circle is divided into \(n\) sectors in such a way that the angles of the sectors are in arithmetic progression. The smallest two angles are \(3 ^ { \circ }\) and \(5 ^ { \circ }\). Find the value of \(n\).

5 The first term of a geometric progression is \(5 \frac { 1 } { 3 }\) and the fourth term is \(2 \frac { 1 } { 4 }\). Find

1. the common ratio,
2. the sum to infinity.

9

1. In an arithmetic progression the sum of the first ten terms is 400 and the sum of the next ten terms is 1000 . Find the common difference and the first term.
2. A geometric progression has first term \(a\), common ratio \(r\) and sum to infinity 6. A second geometric progression has first term \(2 a\), common ratio \(r ^ { 2 }\) and sum to infinity 7 . Find the values of \(a\) and \(r\).

7

1. An athlete runs the first mile of a marathon in 5 minutes. His speed reduces in such a way that each mile takes 12 seconds longer than the preceding mile.
   1. Given that the \(n\)th mile takes 9 minutes, find the value of \(n\).
   2. Assuming that the length of the marathon is 26 miles, find the total time, in hours and minutes, to complete the marathon.
2. The second and third terms of a geometric progression are 48 and 32 respectively. Find the sum to infinity of the progression.

5

1. In a geometric progression, the sum to infinity is equal to eight times the first term. Find the common ratio.
2. In an arithmetic progression, the fifth term is 197 and the sum of the first ten terms is 2040. Find the common difference.

7

1. A geometric progression has first term \(a ( a \neq 0 )\), common ratio \(r\) and sum to infinity \(S\). A second geometric progression has first term \(a\), common ratio \(2 r\) and sum to infinity \(3 S\). Find the value of \(r\).
2. An arithmetic progression has first term 7. The \(n\)th term is 84 and the ( \(3 n\) )th term is 245 . Find the value of \(n\).

8 The first term of a progression is \(4 x\) and the second term is \(x ^ { 2 }\).

1. For the case where the progression is arithmetic with a common difference of 12 , find the possible values of \(x\) and the corresponding values of the third term.
2. For the case where the progression is geometric with a sum to infinity of 8 , find the third term.

6 A ball is such that when it is dropped from a height of 1 metre it bounces vertically from the ground to a height of 0.96 metres. It continues to bounce on the ground and each time the height the ball reaches is reduced. Two different models, \(A\) and \(B\), describe this. Model A: The height reached is reduced by 0.04 metres each time the ball bounces.
Model B: The height reached is reduced by \(4 \%\) each time the ball bounces.

1. Find the total distance travelled vertically (up and down) by the ball from the 1st time it hits the ground until it hits the ground for the 21st time,
   1. using model \(A\),
   2. using model \(B\).
   3. Show that, under model \(B\), even if there is no limit to the number of times the ball bounces, the total vertical distance travelled after the first time it hits the ground cannot exceed 48 metres.