Questions — OCR MEI (4301 questions)

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OCR MEI C2 Q4
5 marks Standard +0.3
4 The second term of a geometric sequence is 6 and the fifth term is - 48 .
Find the tenth term of the sequence.
Find also, in simplified form, an expression for the sum of the first \(n\) terms of this sequence.
OCR MEI C2 Q5
5 marks Moderate -0.8
5 The third term of an arithmetic progression is 24 . The tenth term is 3.
Find the first term and the common difference.
Find also the sum of the 21 st to 50 th terms inclusive. Simplify
  1. Find the 51st term of the sequence given by $$\begin{aligned} u _ { 1 } & = 5 \\ u _ { n + 1 } & = u _ { n } + 4 \end{aligned}$$
  2. Find the sum to infinity of the geometric progression which begins $$5 \quad 2 \quad 0.8 \quad \ldots .$$
OCR MEI C2 Q7
5 marks Moderate -0.8
7 An arithmetic progression has first term 7 and third term 12.
  1. Find the 20th term of this progression.
  2. Find the sum of the 21st to the 50 th terms inclusive of this progression.
OCR MEI C2 Q9
3 marks Moderate -0.8
9 A geometric progression has 6 as its first term. Its sum to infinity is 5 .
Calculate its common ratio.
OCR MEI C2 Q1
5 marks Standard +0.3
1 The 7th term of an arithmetic progression is 6. The sum of the first 10 terms of the progression is 30. Find the 5th term of the progression.
OCR MEI C2 Q2
5 marks Easy -1.2
2 The first three terms of a geometric progression are 4, 2, 1.
Find the twentieth term, expressing your answer as a power of 2 .
Find also the sum to infinity of this progression.
OCR MEI C2 Q3
5 marks Easy -1.2
3 A sequence is given by $$\begin{gathered} a _ { 1 } = 4 \\ a _ { r + 1 } = a _ { r } + 3 \end{gathered}$$ Write down the first 4 terms of this sequence.
Find the sum of the first 100 terms of the sequence.
OCR MEI C2 Q4
11 marks Standard +0.3
4 There is a flowerhead at the end of each stem of an oleander plant. The next year, each flowerhead is replaced by three stems and flowerheads, as shown in Fig. 11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f291e6e3-975e-4d1e-aab6-67308f305da2-2_517_1116_356_455} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure}
  1. How many flowerheads are there in year 5 ?
  2. How many flowerheads are there in year \(n\) ?
  3. As shown in Fig. 11, the total number of stems in year 2 is 4, (that is, 1 old one and 3 new ones). Similarly, the total number of stems in year 3 is 13 , (that is, \(1 + 3 + 9\) ). Show that the total number of stems in year \(n\) is given by \(\frac { 3 ^ { n } - 1 } { 2 }\).
  4. Kitty's oleander has a total of 364 stems. Find
    (A) its age,
    (B) how many flowerheads it has.
  5. Abdul's oleander has over 900 flowerheads. Show that its age, \(y\) years, satisfies the inequality \(y > \frac { \log _ { 10 } 900 } { \log _ { 10 } 3 } + 1\).
    Find the smallest integer value of \(y\) for which this is true.
OCR MEI C2 Q1
3 marks Moderate -0.8
1 A sequence is defined by \(u _ { 1 } = 2\) and \(u _ { k + 1 } = \frac { 10 } { u _ { k } ^ { 2 } }\).
Calculate \(\sum _ { k = 1 } ^ { 4 } u _ { k }\).
OCR MEI C2 Q2
5 marks Easy -1.2
2 The \(n\)th term of a sequence, \(u _ { n }\), is given by $$u _ { n } = 12 - \frac { 1 } { 2 } n .$$
  1. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\). State what type of sequence this is.
  2. Find \(\sum _ { n = 1 } ^ { 30 } u _ { n }\).
OCR MEI C2 Q3
3 marks Moderate -0.8
3 A sequence is defined by $$\begin{aligned} u _ { 1 } & = 10 \\ u _ { r + 1 } & = \frac { 5 } { u _ { r } ^ { 2 } } \end{aligned}$$ Calculate the values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\).
What happens to the terms of the sequence as \(r\) tends to infinity?
OCR MEI C2 Q4
2 marks Moderate -0.8
4 The \(n\)th term, \(t _ { n }\), of a sequence is given by $$t _ { n } = \sin ( \theta + 180 n ) ^ { \circ }$$ Express \(t _ { 1 }\) and \(t _ { 2 }\) in terms of \(\sin \theta ^ { \circ }\).
OCR MEI C2 Q5
13 marks Moderate -0.8
5 Jim and Mary are each planning monthly repayments for money they want to borrow.
  1. Jim's first payment is \(\pounds 500\), and he plans to pay \(\pounds 10\) less each month, so that his second payment is \(\pounds 490\), his third is \(\pounds 480\), and so on.
    (A) Calculate his 12th payment.
    (B) He plans to make 24 payments altogether. Show that he pays \(\pounds 9240\) in total.
  2. Mary's first payment is \(\pounds 460\) and she plans to pay \(2 \%\) less each month than the previous month, so that her second payment is \(\pounds 450.80\), her third is \(\pounds 441.784\), and so on.
    (A) Calculate her 12th payment.
    (B) Show that Jim's 20th payment is less than Mary's 20th payment but that his 19th is not less than her 19th.
    (C) Mary plans to make 24 payments altogether. Calculate how much she pays in total.
    (D) How much would Mary's first payment need to be if she wishes to pay \(2 \%\) less each month as before, but to pay the same in total as Jim, \(\pounds 9240\), over the 24 months?
OCR MEI C2 Q6
2 marks Easy -1.2
6 You are given that $$\begin{aligned} u _ { 1 } & = 1 \\ u _ { n + 1 } & = \frac { u _ { n } } { 1 + u _ { n } } \end{aligned}$$ Find the values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\). Give your answers as fractions.
OCR MEI C2 Q7
4 marks Easy -1.3
7
  1. Evaluate \(\sum _ { r = 2 } ^ { 5 } \frac { 1 } { r - 1 }\).
  2. Express the series \(2 \times 3 + 3 \times 4 + 4 \times 5 + 5 \times 6 + 6 \times 7\) in the form \(\sum _ { r = 2 } ^ { a } \mathrm { f } ( r )\) where \(\mathrm { f } ( r )\) and \(a\) are to be determined.
OCR MEI C2 Q8
3 marks Moderate -0.8
8
  1. Find \(\sum _ { k = 3 } ^ { 8 } \left( k ^ { 2 } - 1 \right)\).
  2. State whether the sequence with \(k\) th term \(k ^ { 2 } - 1\) is convergent or divergent, giving a reason for your answer.
OCR MEI C2 Q9
4 marks Easy -1.2
9
  1. Find the second and third terms of the sequence defined by the following. $$\begin{aligned} t _ { n + 1 } & = 2 t _ { n } + 5 \\ t _ { 1 } & = 3 \end{aligned}$$
  2. Find \(\sum _ { k = 1 } ^ { 3 } k ( k + 1 )\).
OCR MEI C2 Q10
3 marks Easy -1.2
10 For each of the following sequences, state with a reason whether it is convergent, periodic or neither. Each sequence continues in the pattern established by the given terms.
  1. \(3 , \frac { 3 } { 2 } , \frac { 3 } { 4 } , \frac { 3 } { 8 } , \ldots\)
  2. \(3,7,11,15 , \ldots\)
  3. \(3,5 , - 3 , - 5,3,5 , - 3 , - 5 , \ldots\)
OCR MEI C2 Q11
2 marks Easy -1.2
11 Find \(\sum _ { r = 3 } ^ { 6 } r ( r + 2 )\).
OCR MEI C2 Q12
5 marks Moderate -0.8
12 Calculate \(\sum _ { r = 3 } ^ { 6 } \frac { 12 } { r }\). 12 A sequence begins $$\begin{array} { l l l l l l l l l l l } 1 & 3 & 5 & 3 & 1 & 3 & 5 & 3 & 1 & 3 & \ldots \end{array}$$ and continues in this pattern.
  1. Find the 55th term of this sequence, showing your method.
  2. Find the sum of the first 55 terms of the sequence.
OCR MEI C2 Q1
2 marks Easy -1.2
1 Find \(\sum _ { k = 1 } ^ { 5 } \frac { 1 } { 1 + k }\).
OCR MEI C2 Q2
5 marks Moderate -0.8
2 The terms of a sequence are given by $$\begin{aligned} u _ { 1 } & = 192 , \\ u _ { n + 1 } & = - \frac { 1 } { 2 } u _ { n } . \end{aligned}$$
  1. Find the third term of this sequence and state what type of sequence it is.
  2. Show that the series \(u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots\) converges and find its sum to infinity.
OCR MEI C2 Q3
3 marks Easy -1.8
3 A sequence begins $$\begin{array} { l l l l l l l l l l l l } 1 & 2 & 3 & 4 & 5 & 1 & 2 & 3 & 4 & 5 & 1 & \ldots \end{array}$$ and continues in this pattern.
  1. Find the 48th term of this sequence.
  2. Find the sum of the first 48 terms of this sequence.
OCR MEI C2 Q4
3 marks Moderate -0.8
4 Sequences A, B and C are shown below. They each continue in the pattern established by the given terms.
A:1,2,4,32,\(\ldots\)
B:20,- 10,5,- 2.5,1.25,- 0.625,\(\ldots\)
C:20,5,1,20,5,\(\ldots\)
  1. Which of these sequences is periodic?
  2. Which of these sequences is convergent?
  3. Find, in terms of \(n\), the \(n\)th term of sequence A .
OCR MEI C2 Q5
2 marks Easy -1.8
5 Find the numerical value of \(\sum _ { k = 2 } ^ { 5 } k ^ { 3 }\).