1998_Paper II - Section A Pure Mathematics 1 Show that if n...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Section A: Pure Mathematics 1 Show that, if n is an integer such that ( n- 3) 3 + n 3 = ( n + 3) 3 , ( * ) then n is even and n 2 is a factor of 54. Deduce that there is no integer n which satisfies the equation ( * ). Show that, if n is an integer such that ( n- 6) 3 + n 3 = ( n + 6) 3 , ( ** ) then n is even. Deduce that there is no integer n which satisfies the equation ( ** ). 2 Use the first four terms of the binomial expansion of (1- 1 / 50) 1 / 2 , writing 1 / 50 = 2 / 100 to simplify the calculation, to derive the approximation 2 1 . 414214. Calculate similarly an approximation to the cube root of 2 to six decimal places by considering (1 + N/ 125) a , where a and N are suitable numbers. [You need not justify the accuracy of your approximations.] 3 Show that the sum S N of the first N terms of the series 1 1 . 2 . 3 + 3 2 . 3 . 4 + 5 3 . 4 . 5 + + 2 n- 1 n ( n + 1)( n + 2) + is 1 2 3 2 + 1 N + 1- 5 N + 2 . What is the limit of S N as N ? The numbers a n are such that a n a n- 1 = ( n- 1)(2 n- 1) ( n + 2)(2 n- 3) . Find an expression for a n /a 1 and hence, or otherwise, evaluate n =1 a n when a 1 = 2 9 . 1 4 The integral I n is defined by I n = Z ( / 2- x ) sin( nx + x/ 2) cosec ( x/ 2) d x, where n is a positive integer. Evaluate I n- I n- 1 , and hence evaluate I n leaving your answer in the form of a sum. 5 Define the modulus of a complex number z and give the geometric interpretation of | z 1- z 2 | for two complex numbers z 1 and z 2 . On the basis of this interpretation establish the inequality | z 1 + z 2 | 6 | z 1 | + | z 2 | ....
View Full Document

This note was uploaded on 04/01/2012 for the course MATH 1016 taught by Professor Rotar during the Spring '12 term at Central Lancashire.

Page1 / 6

1998_Paper II - Section A Pure Mathematics 1 Show that if n...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online