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2 9. (a) A geometric series has first term a ad commo ratio r. Prove that the sum of the first terms of the series is a(1 r ). 1 r (4) Mr. Kig will be paid a salary of i the year Mr. Kig s cotract promises a 4% icrease i salary every year, the first icrease beig give i 2006, so that his aual salaries form a geometric sequece. (b) Fid, to the earest 100, Mr. Kig s salary i the year Mr. Kig will receive a salary each year from 2005 util he retires at the ed of (c) Fid, to the earest 1000, the total amout of salary he will receive i the period from 2005 util he retires at the ed of (4) 20 *N23492B02028*

3 Questio 9 cotiued *N23492B02128* 21 Tur over

4 4. The first term of a geometric series is 120. The sum to ifiity of the series is (a) Show that the commo ratio, r, is. 4 (b) Fid, to 2 decimal places, the differece betwee the 5th ad 6th term. (c) Calculate the sum of the first 7 terms. The sum of the first terms of the series is greater tha 300. (d) Calculate the smallest possible value of. (3) (4) 8 *N23552A0820*

5 9. A geometric series has first term a ad commo ratio r. The secod term of the series is 4 ad the sum to ifiity of the series is 25. (a) Show that 25r 2 25r + 4 = 0. (b) Fid the two possible values of r. (c) Fid the correspodig two possible values of a. (4) (d) Show that the sum, S, of the first terms of the series is give by S = 25(1 r ). (1) Give that r takes the larger of its two possible values, (e) fid the smallest value of for which S exceeds *N23558A01620*

6 Questio 9 cotiued Q9 (Total 11 marks) *N23558A01720* 17 Tur over

7 2. (a) Fid the first 4 terms, i ascedig powers of x, of the biomial expasio of (1 2x) 5. Give each term i its simplest form. (4) (b) If x is small, so that x 2 ad higher powers ca be igored, show that (1+ x)(1 2x) 5 1 9x. 4 *N24322A0424*

8 10. A geometric series is a + ar + ar (a) Prove that the sum of the first terms of this series is give by (b) Fid S a(1 r ). 1 r (4) (c) Fid the sum to ifiity of the geometric series 10 k 1 k 100(2 ) (3) (3) (d) State the coditio for a ifiite geometric series with commo ratio r to be coverget. (1) 22 *N24322A02224*

9 Questio 10 cotiued (Total 11 marks) TOTAL FOR PAPER: 75 MARKS Q10 END 24 *N24322A02424*

10 physicsadmathstutor.com Jue A tradig compay made a profit of i 2006 (Year 1). A model for future tradig predicts that profits will icrease year by year i a geometric sequece with commo ratio r, r > 1. The model therefore predicts that i 2007 (Year 2) a profit of r will be made. (a) Write dow a expressio for the predicted profit i Year. The model predicts that i Year, the profit made will exceed log 4 (b) Show that > + 1. log r (1) (3) Usig the model with r = 1.09, (c) fid the year i which the profit made will first exceed , (d) fid the total of the profits that will be made by the compay over the 10 years from 2006 to 2015 iclusive, givig your aswer to the earest (3) 18 *H26108A01824*

11 physicsadmathstutor.com Jue 2007 Questio 8 cotiued Q8 (Total 9 marks) *H26108A01924* 19 Tur over

12 2. The fourth term of a geometric series is 10 ad the seveth term of the series is 80. For this series, fid (a) the commo ratio, (b) the first term, (c) the sum of the first 20 terms, givig your aswer to the earest whole umber. 4 *H26320B0424*

13 6. A geometric series has first term 5 ad commo ratio 4. 5 Calculate (a) the 20th term of the series, to 3 decimal places, (b) the sum to ifiity of the series. Give that the sum to k terms of the series is greater tha 24.95, log (c) show that k >, log 0.8 (4) (d) fid the smallest possible value of k. (1) 12 *H30722A01228*

14 Questio 6 cotiued *H30722A01328* 13 Tur over

15 9. The first three terms of a geometric series are (k + 4), k ad (2k 15) respectively, where k is a positive costat. (a) Show that k 2 7k 60 = 0. (b) Hece show that k = 12. (c) Fid the commo ratio of this series. (4) (d) Fid the sum to ifiity of this series. 22 *H30957A02228*

16 Questio 9 cotiued *H30957A02328* 23 Tur over

17 5. The third term of a geometric sequece is 324 ad the sixth term is 96 (a) Show that the commo ratio of the sequece is 2 3 (b) Fid the first term of the sequece. (c) Fid the sum of the first 15 terms of the sequece. (3) (d) Fid the sum to ifiity of the sequece. 12 *H34263A01224*

18 Questio 5 cotiued Q5 (Total 9 marks) *H34263A01324* 13 Tur over

19 6. A car was purchased for o 1st Jauary. 1st Jauary each followig year, the value of the car is 80 of its value o 1st Jauary i the previous year. (a) Show that the value of the car exactly 3 years after it was purchased is (1) The value of the car falls below 1000 for the first time (b) Fid the value of. years after it was purchased. (3) A isurace compay has a scheme to cover the maiteace of the car. The cost is 200 for the first year, ad for every followig year the cost icreases by 12 so that for the 3rd year the cost of the scheme is (c) Fid the cost of the scheme for the 5th year, givig your aswer to the earest pey. (d) Fid the total cost of the isurace scheme for the first 15 years. (3) 10 *N35101A01024*

20 Questio 6 cotiued *N35101A01124* 11 Tur over

21 9. The adult populatio of a tow is at the ed of Year 1. A model predicts that the adult populatio of the tow will icrease by 3% each year, formig a geometric sequece. (a) Show that the predicted adult populatio at the ed of Year 2 is (b) Write dow the commo ratio of the geometric sequece. (1) (1) The model predicts that Year tow exceeds will be the first year i which the adult populatio of the (c) Show that ( 1) log1.03> log1.6 (3) (d) Fid the value of. At the ed of each year, each member of the adult populatio of the tow will give 1 to a charity fud. Assumig the populatio model, (e) fid the total amout that will be give to the charity fud for the 10 years from the ed of Year 1 to the ed of Year 10, givig your aswer to the earest (3) 22 *H35384A02228*

22 Questio 9 cotiued *H35384A02328* 23 Tur over

23 3. The secod ad fifth terms of a geometric series are 750 ad 6 respectively. Fid (a) the commo ratio of the series, (b) the first term of the series, (3) (c) the sum to ifiity of the series. 6 *H35403A0628*

24 6. The secod ad third terms of a geometric series are 192 ad 144 respectively. For this series, fid (a) the commo ratio, (b) the first term, (c) the sum to ifiity, (d) the smallest value of for which the sum of the first terms of the series exceeds (4) 16 *P38158A01632*

25 Questio 6 cotiued *P38158A01732* 17 Tur over

26 1. A geometric series has first term a = 360 ad commo ratio r = 7 8 Givig your aswers to 3 sigificat figures where appropriate, fid (a) the 20th term of the series, (b) the sum of the first 20 terms of the series, (c) the sum to ifiity of the series. 2 *P40083A0228*

27 9. A geometric series is a + ar + ar (a) Prove that the sum of the first terms of this series is give by S a( 1 r ) = 1 r (4) The third ad fifth terms of a geometric series are 5.4 ad respectively ad all the terms i the series are positive. For this series fid, (b) the commo ratio, (c) the first term, (d) the sum to ifiity. (3) 26 *P40685A02628*

28 Questio 9 cotiued Q9 (Total 11 marks) TOTAL FOR PAPER: 75 MARKS END 28 *P40685A02828*

29 3. A compay predicts a yearly profit of i the year The compay predicts that the yearly profit will rise each year by 5%. The predicted yearly profit forms a geometric sequece with commo ratio 1.05 (a) Show that the predicted profit i the year 2016 is (b) Fid the first year i which the yearly predicted profit exceeds (1) (5) (c) Fid the total predicted profit for the years 2013 to 2023 iclusive, givig your aswer to the earest poud. (3) 6 *P41487A0632*

30 physicsadmathstutor.com Jue 2013 (R) 5. The first three terms of a geometric series are 4p, (3p + 15) ad (5p + 20) respectively, where p is a positive costat. (a) Show that 11p 2 10p 225 = 0 (b) Hece show that p = 5 (c) Fid the commo ratio of this series. (4) (d) Fid the sum of the first te terms of the series, givig your aswer to the earest iteger. (3) 12 *P42826A01232*

31 physicsadmathstutor.com Jue 2013 (R) Questio 5 cotiued *P42826A01332* 13 Tur over

32 physicsadmathstutor.com Jue The first three terms of a geometric series are respectively, where p is a costat. 18, 12 ad p Fid (a) the value of the commo ratio of the series, (b) the value of p, (1) (1) (c) the sum of the first 15 terms of the series, givig your aswer to 3 decimal places. 2 *P41859A0232*

33 Edexcel AS/A level Mathematics Formulae List: Core Mathematics C2 Issue 1 September Core Mathematics C2 Cadidates sittig C2 may also require those formulae listed uder Core Mathematics C1. Cosie rule a 2 = b 2 + c 2 2bc cos A Biomial series 2 1 ) ( r r b b a r b a b a a b a = + K K ( N) where )!!(! C r r r r = = < = + x x r r x x x r 1, ( 2 1 1) ( 1) ( 2 1 1) ( 1 ) (1 2 K K K K R) Logarithms ad expoetials a x x b b a log log log = Geometric series u = ar 1 S = r r a 1 ) (1 S = r a 1 for r < 1 Numerical itegratio The trapezium rule: b a x y d 21 h{(y 0 + y ) + 2(y 1 + y y 1 )}, where a b h =

34 Core Mathematics C1 Mesuratio Surface area of sphere = 4π r 2 Area of curved surface of coe = π r slat height Arithmetic series u = a + ( 1)d S = 2 1 (a + l) = 2 1 [2a + ( 1)d] 4 Edexcel AS/A level Mathematics Formulae List: Core Mathematics C1 Issue 1 September 2009

Further Pure 1 Revision Topic 5: Sums of Series

Further Pure 1 Revision Topic 5: Sums of Series The OCR syllabus says that cadidates should: Further Pure Revisio Topic 5: Sums of Series Cadidates should be able to: (a) use the stadard results for Σr, Σr, Σr to fid related sums; (b) use the method