MATHS 9233 /

REVISION TUTORIAL 1

/ YR1/2003

PARTIAL FRACTIONS

  1. Express in partial fractions .
  2. Express in partial fractions .
  3. Express in partial fractions .

BINOMIAL EXPANSION

1. Find the values of A, B and C to satisfy the identity .

Hence, or otherwise, obtain the expansion of in ascending powers of x, giving the first five terms of the expansion. State the set of values of x for which the expansion is valid.

[8, -3, 1; 1 +3x + 7x2 - 3x2/2 - 47x4/4; |x| < 1/2]

  1. Given that |x| < 2, expand as a series in ascending powers of x up to and including the term

in x4. By using x = 2/3, find the value of 8 to three decimal places. [2(1- x2/8 - x4/128); 2.829]

  1. Expand in ascending powers of x up to and including the term in x3, simplifying the

coefficients. By taking x = 4/100, use the above expansion to obtain an approximate value for 6,

giving your answer correct to two decimal places. [1 + ½ x + 3x2/8 + 5x3/16 + …; 2.45]

  1. Obtain the expansion of in the form x2/2 + ax3 + bx4 + cx5 + dx6 + … where a, b, c and d are

to be determined. State the range of values of x for which the expansion is valid.

[ a= 0, b = 1/16 , c = 0, d =3/256; |x| < 2]

  1. Obtain an expansion of in ascending powers of 1/x as far as the term in 1/x3. State the set of values of x for which this expansion is valid. [2-1/3(1 - 1/6x + 1/18x2 - 7/324x3 +…; |x| > ½]
  1. Show that, if x is so small that terms in x3 and higher powers may be neglected, then . By putting x = 1/9, show that 5 is approximately equal to 181/81.
  2. Write down the first three non-zero terms in the expansion of in descending powers of x. State the range of x for the expansion to be valid. Hence obtain an approximate value of 26 to 3 decimal places. [x + 1/2x - 1/8x3 +…;|x|>1; 5.099]

8(i) Obtain the binomial expansion for |x| < 1, of in terms of n, where n is a constant, up to and

including the term in x2. Let x = 1/15. Find an approximation to the fifth root of 7/8, leaving your answer as a fraction. [1 - 2nx + 2n2x2 + …; 5477/5625]

(ii) Expand (1 + x + ax2)3 in ascending powers of x as far as the term in x2. [1+ 3x + 3(a + 1)x2 +…]

(iii) Hence find the values of a and n if . [a = ½ ,n = -3/2]

ARITHMETIC AND GEOMETRIC PROGRESSIONS

1. The sum of the first n terms of a series is n2(n + 1) for all positive integers n.

(a) Determine the nth term. [n(3n - 1)]

(b) Calculate the sum from the 5th to the 23rd term. [12616]

  1. Find the sum to infinity of the series 1 + 2 cos2 + 4cos4 + …. stating clearly the values of  for which the sum exists. [45o < 135o, 225o < 315o]
  1. The nth term of an infinite geometric series is equal to one fourth of the sum of all the terms after (but not including) the nth term. Show that the sum to infinity of the geometric series is five times its first term.
  1. The first two terms of a geometric progression are a and b ( b < a). If the sum of the first n terms is equal to the sum to infinity of the remaining terms, prove that an = 2bn. If a = 1 and b = 1/3, write down an expression for Sn, the sum of the first n terms. Find S, the sum to infinity and the least value of n for which S - Sn < 0.01. [1.5(1 - (1/3)n) ; 1.5 ; 5]
  1. A nail 4 cm long is driven into wood by blows of a hammer. The first blow drives it 2.5 cm and each

successive blow drives it in two fifths of the previous distance (except the last for which the

distance is less). Find how many blows must be used. [4]

  1. The second term of a geometric progression is 8 and the sum to infinity is 32. Find the first term and the common ratio. Show that the sum of all terms after the nth term is 2 5-n and find the least value of n such that this sum is less than 2% of the sum to infinity. [a = 16, r = ½, n = 6]
  2. If a, b are two different real numbers, show . Given that a1, a2, a3,……,an is an

arithmetic progression with common difference d, show that

.

8(a) Let n denotes the nth term of an increasing arithmetic progression such that 1, 4 and 8 are in

geometric progression and 10 + 12 + 14 +…+ 30 = 924. Find the 1st term and common difference of the arithmetic progression. [27; 3]

(b)Given that Tn = log10(pqn-1) where p and q are positive constants, show that the terms T1, T2, T3,

….Tn form an arithmetic progression. Find an expression for Sn, the sum to n terms. Hence obtain a relationship between p and q if S3 = 3. [ nlog10p + [n(n - 1)log10q]/2 ; p = 10/q]

SUMMATION NOTATION

  1. Express in terms of n. Hence find the value of

2. Given that [1/4]

  1. Given that Hence, find or otherwise, determine the sum of the series
  1. Show that the solution of N to the equation must satisfy the equation

(2N + 1)(N + 1)(N - 6) = 30.

  1. Let Find the range of values of x so that the series A converges.

Given that x = 0, find Use your answer to write down the value of A. State whether

is greater or smaller than A.

6. Show that [1814]

7. Express in terms of n, where n>0.

MATHEMATICAL INDUCTION

  1. Prove by mathematical induction that for all positive integers n.
  1. Prove by mathematical induction that, for every positive integers n,
  1. Prove by mathematical induction that

4. Use mathematical induction to prove that the sum to n terms of the series

  1. Show by mathematical induction that, for all positive integers n,
  1. Show by induction that, for every positive integers n,
  1. A sequence of numbers for all positive

integral values of n. Prove by induction that

8. Show by induction that, for all positive integers n,

INEQUALITIES

  1. Find the range of values of x for . [
  2. Sketch the graph of . Hence, deduce the set of values of x for which .
  3. Find the values of x for which . ]
  4. (a) Solve the inequality .

(b) Find the values of x such that | 2x | = | x – 1 |

By sketching the graphs of and on the same axes, find the values of x

such that . Hence find the values of x such that . []

  1. Solve the inequality . []
  2. Solve the inequalities (i) . (ii) .

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