Sequences and series assessment

Transcription

1 Red ) a) Find the sum of all the integers between and 000 which are divisible by 7 [3] b) 42 Hence, or otherwise, evaluate (7r + 2) r= [2] 2) The first three terms of an arithmetic series are k, 7.5, and k + 7 a) Find the value of k b) Find the sum of the first eleven terms of the series [3] [2] 3) A sequence of positive integers, u u 2 u 3, is given by u =, and u n+ = 7 u n a) Write down the first four terms of the sequence [3] b) Express u n in terms of n [2]

2 4) The sum of an arithmetic series is n r= ( 80 r ) a) Write down the first two terms of the series. [2] b) Find the common difference of the series. [] c) Given that n = 36, find the sum of the series. [3] Amber 5) The first three terms of an arithmetic series are p, 5p 8, and 3p + 8 respectively a) Show that p = 4 [3] b) Find the 8 th term of the sequence [2] 2

3 6) The sequence u, u2, u3,, un is defined by the recurrence relation un + = pun + 5, u = 2, where p is a constant a) Given that u3 = 8, show that one possible value of p is 2 and find the other value of p [4] b) Find both possible values of u4 [2] 7) In the first month after opening, a mobile phone shop sold 280 phones. A model for future trading assumes that sales will increase by x phones per month for the next 35 months, so that (280 + x) phones will be sold in the second month, ( x) in the third month, and so on. Using this model with x = 5 find a) the number of phones sold in the 36th month [3] b) the total number of phones sold over the 36 months [3] The shop sets a sales target of phones to be sold over the 36 months c) Using the same model, find the least value of x required to achieve this target [4] 3

4 8) A geometric series is a + ar + ar 2 + a) Find, to 3 significant figures, the sum of the first twenty terms of G [3] b) Find the sum to infinity of G [2] Another geometric series has its first term equal to its common ratio. The sum to infinity of this series is 0. c) Find the exact value of the common ratio of this series. [3] 9) A company made a profit of in the year 200. A model for future performance assumes that yearly profits will increase in an arithmetic sequence with common difference d. This model predicts total profits of for the 9 years 200 to 2009 inclusive. a) Find the value of d [3] b) Using your value of d, find the predicted profit for the year 20 c) An alternative model assumes that the company s yearly profits will increase in a geometric sequence with common ratio.06. Using this alternative model and again taking the profit in 200 to be , find the predicted profit for the year 20 [2] [2] 4

5 0) The sum of an arithmetic series is: a) Write down the first two terms of the series [2] b) Find the common difference of the series [] c) Given that n = 50, find the sum of the series [3] n r= (80 3r) Green ) A company made a profit of in the year 200. A model for future performance assumes that yearly profits will increase in an arithmetic sequence with common difference d. This model predicts total profits of for the 9 years 200 to 2009 inclusive a) Find the value of d [4] b) Using your value of d, find the predicted profit for the year 20 [3] 5

6 2) Show that n r= (4r + 5) = n(2n + 7) [3] 3) A geometric series has first term 200. Its sum to infinity is 960. a) Show that the common ratio of the series is 4. [2] b) Find, to 3 decimal places, the difference between the ninth and tenth terms of the series. [3] c) Write down an expression for the sum of the first n terms of the series. [] d) Given that n is odd, prove that the sum of the first n terms of the series is 960( n ). [2] 6

7 4) In a Geometric series the common ratio is r and sum to n terms is S n Given is S = 8 7 is S 6 Show that is r = ± k, where is k is an integer to be found [4] 5) The second term of a geometric series is 2. and the fifth term of the series is a) Find the first term and the common ratio of the series [4] b) Find the sum to infinity of the series as an exact fraction. [2] c) Find the difference between the sum to infinity of the series and the sum of the first 4 terms, giving your answer in the form a 0 n, where n < 0 and n is an integer [4] 7

8 6) The sum to infinity of a geometric series with first term a and common ratio r is 5. The sum to infinity of a second geometric series with first term a and common ratio 4r is 2 a) Find the value of r. [3] b) Find the sum of the first 4 terms of the first series, giving the exact answer [3] 8

9 TOTAL 0 A B C D E 80% 70% 60% 50% 40% WWW: EBI: 9