1.04i Geometric sequences: nth term and finite series sum

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 \(P\).

2

1. Find, in terms of the non-zero constant \(k\), the first 4 terms in the expansion of \(( k + x ) ^ { 8 }\) in ascending powers of \(x\).
2. Given that the coefficients of \(x ^ { 2 }\) and \(x ^ { 3 }\) in this expansion are equal, find the value of \(k\).

5 \includegraphics[max width=\textwidth, alt={}, center]{73c0c113-8f35-4e7f-ad5d-604602498b71-2_741_533_1279_808} The diagram shows a pyramid \(O A B C\) with a horizontal base \(O A B\) where \(O A = 6 \mathrm {~cm} , O B = 8 \mathrm {~cm}\) and angle \(A O B = 90 ^ { \circ }\). The point \(C\) is vertically above \(O\) and \(O C = 10 \mathrm {~cm}\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O B\) and \(O C\) as shown. Use a scalar product to find angle \(A C B\).

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 \(S\).

10

1. An arithmetic progression contains 25 terms and the first term is - 15 . The sum of all the terms in the progression is 525. Calculate
   1. the common difference of the progression,
   2. the last term in the progression,
   3. the sum of all the positive terms in the progression.
2. A college agrees a sponsorship deal in which grants will be received each year for sports equipment. This grant will be \(\\) 4000\( in 2012 and will increase by \)5 \%$ each year. Calculate
   1. the value of the grant in 2022,
   2. the total amount the college will receive in the years 2012 to 2022 inclusive.

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.

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.

5 The sum of the 1st and 2nd terms of a geometric progression is 50 and the sum of the 2nd and 3rd terms is 30 . Find the sum to infinity.

8

1. A cyclist completes a long-distance charity event across Africa. The total distance is 3050 km . He starts the event on May 1st and cycles 200 km on that day. On each subsequent day he reduces the distance cycled by 5 km .
   1. How far will he travel on May 15th?
   2. On what date will he finish the event?
2. A geometric progression is such that the third term is 8 times the sixth term, and the sum of the first six terms is \(31 \frac { 1 } { 2 }\). Find
   1. the first term of the progression,
   2. the sum to infinity of the progression.

1 An arithmetic progression has first term - 12 and common difference 6 . The sum of the first \(n\) terms exceeds 3000 . Calculate the least possible value of \(n\).

9 Relative to an origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 8 \\ - 6 \\ 5 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } - 10 \\ 3 \\ - 13 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 2 \\ - 3 \\ - 1 \end{array} \right)$$ A fourth point, \(D\), is such that the magnitudes \(| \overrightarrow { A B } | , | \overrightarrow { B C } |\) and \(| \overrightarrow { C D } |\) are the first, second and third terms respectively of a geometric progression.

1. Find the magnitudes \(| \overrightarrow { A B } | , | \overrightarrow { B C } |\) and \(| \overrightarrow { C D } |\).
2. Given that \(D\) is a point lying on the line through \(B\) and \(C\), find the two possible position vectors of the point \(D\).

4 The first term of a series is 6 and the second term is 2 .

1. For the case where the series is an arithmetic progression, find the sum of the first 80 terms.
2. For the case where the series is a geometric progression, find the sum to infinity.

4 A runner who is training for a long-distance race plans to run increasing distances each day for 21 days. She will run \(x \mathrm {~km}\) on day 1 , and on each subsequent day she will increase the distance by \(10 \%\) of the previous day's distance. On day 21 she will run 20 km .

1. Find the distance she must run on day 1 in order to achieve this. Give your answer in km correct to 1 decimal place.
2. Find the total distance she runs over the 21 days.

8

1. Over a 21-day period an athlete prepares for a marathon by increasing the distance she runs each day by 1.2 km . On the first day she runs 13 km .
   1. Find the distance she runs on the last day of the 21-day period.
   2. Find the total distance she runs in the 21-day period.
2. The first, second and third terms of a geometric progression are \(x , x - 3\) and \(x - 5\) respectively.
   1. Find the value of \(x\).
   2. Find the fourth term of the progression.
   3. Find the sum to infinity of the progression.

9 The first, second and third terms of a geometric progression are \(3 k , 5 k - 6\) and \(6 k - 4\), respectively.

1. Show that \(k\) satisfies the equation \(7 k ^ { 2 } - 48 k + 36 = 0\).
2. Find, showing all necessary working, the exact values of the common ratio corresponding to each of the possible values of \(k\).
3. One of these ratios gives a progression which is convergent. Find the sum to infinity.

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.

5

1. State the sum of the series \(z + z ^ { 2 } + z ^ { 3 } + \ldots + z ^ { n }\), for \(z \neq 1\).
2. Given that \(z\) is an \(n\)th root of unity and \(z \neq 1\), deduce that \(1 + z + z ^ { 2 } + \ldots + z ^ { n - 1 } = 0\).
3. Given instead that \(z = \frac { 1 } { 3 } ( \cos \theta + \mathrm { i } \sin \theta )\), use de Moivre's theorem to show that $$\sum _ { m = 1 } ^ { \infty } 3 ^ { - m } \cos m \theta = \frac { 3 \cos \theta - 1 } { 10 - 6 \cos \theta }$$

4 The diagram shows the curve with equation \(\mathrm { y } = 2 ^ { \mathrm { x } }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(N\) rectangles each of width \(\frac { 1 } { N }\). \includegraphics[max width=\textwidth, alt={}, center]{114ece0d-558d-4c02-8a77-034b3681cff9-06_824_1161_376_450}

1. By considering the sum of the areas of these rectangles, show that \(\int _ { 0 } ^ { 1 } 2 ^ { x } d x < U _ { N }\), where $$\mathrm { U } _ { \mathrm { N } } = \frac { 2 ^ { \frac { 1 } { \mathrm {~N} } } } { \mathrm {~N} \left( 2 ^ { \frac { 1 } { \mathrm {~N} } } - 1 \right) }$$
2. Use a similar method to find, in terms of \(N\), a lower bound \(\mathrm { L } _ { \mathrm { N } }\) for \(\int _ { 0 } ^ { 1 } 2 ^ { x } \mathrm {~d} x\).
3. Find the least value of \(N\) such that \(\mathrm { U } _ { \mathrm { N } } - \mathrm { L } _ { \mathrm { N } } < 10 ^ { - 4 }\).