Mathematics

1. 1.

   The 5th term of an arithmetic sequence is 17 and the 12th term is 38. Find the first term, the common difference and the sum of the first 20 terms.

   [3 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. Subtracting the two term equations eliminates a and gives d directly.
   2. Use S_n = (n/2)(2a + (n - 1)d) once a and d are known.

   Marking points

   • Forms 7d = 38 - 17 so d = 3.
   • From a + 4d = 17 obtains a = 5.
   • S20 = (20/2)(2 * 5 + 19 * 3) = 670.

   Examiner tip: The nth term is a + (n - 1)d. Use the sum formula with n - 1 = 19, not 20.

2. 2.

   A geometric sequence begins 3, 6, 12, ... Find the first term that exceeds 1000 and its value.

   [3 marks]

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. Write the general term and form the inequality; test powers of 2 around 333 (2^8 = 256, 2^9 = 512).
   2. Check the neighbouring term to make sure the answer is the first.

   Marking points

   • The nth term is 3 * 2^(n - 1).
   • Solves 3 * 2^(n - 1) > 1000, so 2^(n - 1) > 333.3.
   • n = 10 is the first, and the term is 1536 (the 9th term 768 is too small).

   Examiner tip: Check the term just before as well. Use logarithms if the numbers are not convenient.

3. 3.

   A geometric series has first term 12 and sum to infinity 48. Find the common ratio, the second term and the sum of the first three terms.

   [4 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. Rearrange the infinite-sum formula for r; the series converges because |r| < 1.
   2. List the first three terms (12, 9, 6.75) and add them.

   Marking points

   • Uses S_inf = a/(1 - r): 12/(1 - r) = 48.
   • Obtains r = 3/4.
   • The second term is 12 * 3/4 = 9.
   • S3 = 12 + 9 + 6.75 = 27.75.

   Examiner tip: The sum to infinity only exists when |r| < 1. Always state this condition.

4. 4.

   Expand (1 + 2x)^(-2) in ascending powers of x up to and including the term in x^3, and state the range of values of x for which the expansion is valid.

   [4 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. The general expansion is (1 + u)^n = 1 + nu + n(n - 1)u^2/2! + n(n - 1)(n - 2)u^3/3! + ... for |u| < 1.
   2. Replace u by 2x and bracket the substitution so the powers of 2 are applied.

   Marking points

   • Uses (1 + u)^(-2) = 1 - 2u + 3u^2 - 4u^3 with u = 2x.
   • Substitutes u = 2x correctly: -2(2x) = -4x and 3(2x)^2 = 12x^2.
   • Obtains 1 - 4x + 12x^2 - 32x^3.
   • Valid for |2x| < 1, so |x| < 1/2.

   Examiner tip: Put the whole term 2x in brackets before applying the power, so 2^2 and 2^3 are not forgotten. State |x| < 1/2, not |x| < 1.

5. 5.

   Show that the sum of the first n terms of the series with r-th term (3r - 1) is n(3n + 1)/2. Hence find the least n for which the sum exceeds 500.

   [5 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. Use the standard result for the sum of the first n integers and remember the constant term sums to n.
   2. For the inequality, evaluate the closed form at consecutive values near the solution of n(3n + 1) = 1000.

   Marking points

   • Splits the sum: 3 times the sum of r minus the sum of 1.
   • Uses sum r = n(n + 1)/2 to get 3n(n + 1)/2 - n.
   • Simplifies to (3n^2 + n)/2 = n(3n + 1)/2.
   • Tests n = 18 giving 495, which is below 500.
   • Tests n = 19 giving 551, so the least n is 19.

   Examiner tip: Show the algebra for a 'show that' question. For the least integer, test both n and n - 1.

6. 6.

   (a) Expand (4 + x)^(1/2) as 2(1 + x/4)^(1/2) up to the term in x^2. (b) Use x = 0.2 to estimate the square root of 4.2 to 5 decimal places. (c) State the values of x for which the expansion is valid.

   [5 marks]

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. To use the binomial expansion the first term inside the bracket must be 1, so factor 4 out first, remembering the square root of 4 is 2.
   2. Compare with the calculator value 2.04939 to see how accurate two terms are: the error is under 0.00002.

   Marking points

   • Factors out 4: (4 + x)^(1/2) = 2(1 + x/4)^(1/2).
   • Expands (1 + u)^(1/2) = 1 + u/2 - u^2/8 with u = x/4.
   • Obtains 2 + x/4 - x^2/64.
   • With x = 0.2: 2 + 0.05 - 0.000625 = 2.049375.
   • Valid for |x/4| < 1, so |x| < 4.

   Examiner tip: The 'validity' condition comes from the bracket |x/4| < 1, not |x| < 1.

Original, unofficial practice questions from SubjectScout — not official past-paper questions.