1. For the sequence 18, -9,9/2, -9/4, .....
(a) Write down a formula for the ith term, Ti;
(b) Using (a), or otherwise, find T12. Express your answer as a fraction in simplest form;
(c) Write down using summation notation a formula for the sum of the first n terms. If this formula simplifies, then simplify it;
(d) Compute S12 and S∞. Express S12 correct to three decimal places. Express S∞ exactly as an improper fraction.
2. Using a geometric series, write 0.25252525... as a rational number (i.e. a proper fraction).
3. James puts $8,500 in a bank for superannuation purposes, with an interest rate of 2.5% per year. The interest is added at the end of each year, and the interest rate is applied to the new amount over the next year. After how many years (as a whole number) will James have a total of $35,000 in the superannuation account?
4. In an arithmetic progression, T1, T2, . . . it is known that T6 = 34 and T12 = 56.
(a) Determine a formula for the ith term of this sequence;
(b) Using (a), or otherwise, find the term T15 of the sequence.
5. Bob is working part-time at an annual salary of $17,900. After each year he receives an increase of $950. What is his salary in his 13th year? What are his total earnings over the first 13 years?
6. A bucket contains three pink balls, five green balls and two yellow balls. (a) In total, how many different ways can the balls be drawn out of the bucket? Hint: Here we are not concerned about the colour of the balls; think of each ball as different. (b) How many ways could I draw a pink ball, then a green ball and finally a yellow ball out of the bucket? What is the probability of this occurring?
7. Using an appropriate combination nCr, what is the coefficient of x^6 in the expansion of(3 - 2x)^10? Note: You do not have to write out the full expansion to determine the required coefficient. Be sure to state the name of any theorem you use.
8. A simple, ordinary annuity has $300 paid in each month and provides an interest rate of 0.5% per month. Find (a) The value of the annuity after 10 months; (b) The value of the annuity after 12 years. Note: You may assume the financial institution uses simple rounding to the nearest cent