Mathematics
Sequences, series and the binomial expansion
- 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 - 2.
A geometric sequence begins 3, 6, 12, ... Find the first term that exceeds 1000 and its value.
[3 marks] - 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 - 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 - 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 - 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]
Marking points are indicative, not an official mark scheme. Accept equivalent valid methods and supported interpretations that address the task; award each mark once without requiring the model wording.