Questions — OCR MEI C2 (454 questions)

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OCR MEI C2 Q1
1 An arithmetic progression has tenth term 11.1 and fiftieth term 7.1. Find the first term and the common difference. Find also the sum of the first fifty terms of the progression.
OCR MEI C2 Q2
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. How many of Jill's descendants would there be altogether in the first 15 generations?
  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. 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?
OCR MEI C2 Q3
3
  1. Find \(\sum _ { r = 1 } ^ { 5 } \frac { 21 } { r + 2 }\).
  2. A sequence is defined by $$\begin{aligned} u _ { 1 } & = a , \text { where } a \text { is an unknown constant, }
    u _ { n + 1 } & = u _ { n } + 5 . \end{aligned}$$ Find, in terms of \(a\), the tenth term and the sum of the first ten terms of this sequence.
OCR MEI C2 Q4
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 \(r\).
\(5 \quad S\) is the sum to infinity of a geometric progression with first term \(a\) and common ratio \(r\).
  1. Another geometric progression has first term \(2 a\) and common ratio \(r\). Express the sum to infinity of this progression in terms of \(S\).
  2. A third geometric progression has first term \(a\) and common ratio \(r ^ { 2 }\). Express, in its simplest form, the sum to infinity of this progression in terms of \(S\) and \(r\).
OCR MEI C2 Q6
6 Find the second and third terms in the sequence given by $$\begin{aligned} & u _ { 1 } = 5
& u _ { n + 1 } = u _ { n } + 3 \end{aligned}$$ Find also the sum of the first 50 terms of this sequence.
OCR MEI C2 Q7
7 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 \(r\).
  2. Show that the ratio of the \(n\)th terms of the two geometric progressions found in part (i) can be written as \(2 ^ { n - 2 } : 3 ^ { n - 2 }\).
OCR MEI C2 Q1
1
  1. In a 'Make Ten' quiz game, contestants get \(\pounds 10\) for answering the first question correctly, then a further \(\pounds 20\) for the second question, then a further \(\pounds 30\) for the third, and so on, until they get a question wrong and are out of the game.
    (A) Haroon answers six questions correctly. Show that he receives a total of \(\pounds 210\).
    (B) State, in a simple form, a formula for the total amount received by a contestant who answers \(n\) questions correctly. Hence find the value of \(n\) for a contestant who receives \(\pounds 10350\) from this game.
  2. In a 'Double Your Money' quiz game, contestants get \(\pounds 5\) for answering the first question correctly, then a further \(\pounds 10\) for the second question, then a further \(\pounds 20\) for the third, and so on doubling the amount for each question until they get a question wrong and are out of the game.
    (A) Gary received \(\pounds 75\) from the game. How many questions did he get right?
    (B) Bethan answered 9 questions correctly. How much did she receive from the game?
    (C) State a formula for the total amount received by a contestant who answers \(n\) questions correctly. Hence find the value of \(n\) for a contestant in this game who receives \(\pounds 2621435\).
OCR MEI C2 Q2
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.
  2. Find the sum to infinity of the series.
OCR MEI C2 Q3
3 The 11th term of an arithmetic progression is 1. The sum of the first 10 terms is 120. Find the 4th term.
OCR MEI C2 Q4
4
  1. André is playing a game where he makes piles of counters. He puts 3 counters in the first pile. Each successive pile he makes has 2 more counters in it than the previous one.
    1. How many counters are there in his sixth pile?
    2. André makes ten piles of counters. How many counters has he used altogether?
  2. In another game, played with an ordinary fair die and counters, Betty needs to throw a six to start. The probability \(\mathrm { P } _ { n }\) of Betty starting on her \(n\)th throw is given by $$P _ { n } = \frac { 1 } { 6 } \times \left( \frac { 5 } { 6 } \right) ^ { n - 1 }$$
    1. Calculate \(\mathrm { P } _ { 4 }\). Give your answer as a fraction.
    2. The values \(\mathrm { P } _ { 1 } , \mathrm { P } _ { 2 } , \mathrm { P } _ { 3 } , \ldots\) form an infinite geometric progression. State the first term and the common ratio of this progression. Hence show that \(\mathrm { P } _ { 1 } + \mathrm { P } _ { 2 } + \mathrm { P } _ { 3 } + \ldots = 1\).
    3. Given that \(\mathrm { P } _ { n } < 0.001\), show that \(n\) satisfies the inequality $$n > \frac { \log _ { 10 } 0.006 } { \log _ { 10 } \left( \frac { 5 } { 6 } \right) } + 1$$ Hence find the least value of \(n\) for which \(\mathrm { P } _ { n } < 0.001\).
OCR MEI C2 Q5
5 The first term of a geometric series is 8. The sum to infinity of the series is 10 .
Find the common ratio.
OCR MEI C2 Q1
1
  1. An arithmetic progression has first term \(A\) and common difference \(D\). The sum of its first two terms is 25 and the sum of its first four terms is 250 .
    (A) Find the values of \(A\) and \(D\).
    (B) Find the sum of the 21 st to 50 th terms inclusive of this sequence.
  2. A geometric progression has first term \(a\) and common ratio \(r\), with \(r \neq \pm 1\). The sum of its first two terms is 25 and the sum of its first four terms is 250 . Use the formula for the sum of a geometric progression to show that \(\frac { r ^ { 4 } - 1 } { r ^ { 2 } - 1 } = 10\) and hence or otherwise find algebraically the possible values of \(r\) and the corresponding values of \(a\).
OCR MEI C2 Q2
2 A geometric progression has a positive common ratio. Its first three terms are 32, \(b\) and 12.5.
Find the value of \(b\) and find also the sum of the first 15 terms of the progression.
OCR MEI C2 Q3
3 In an arithmetic progression, the second term is 11 and the sum of the first 40 terms is 3030 . Find the first term and the common difference.
OCR MEI C2 Q4
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 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
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
9 A geometric progression has 6 as its first term. Its sum to infinity is 5 .
Calculate its common ratio.
OCR MEI C2 Q1
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
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
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
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
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
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 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?