Topic 1. Sequences and Series, Exponentials and The Binomial Theorem
1.Find the sum of the arithmetic series
17 + 27 + 37 +...+ 417.
2.Find the coefficient of x5 in the expansion of (3x – 2)8. 3.An arithmetic series has five terms. The first term is 2 and the last term is 32. Find the sum of the series. 4.Find the coefficient of a3b4 in the expansion of (5a + b)7. 5.Solve the equation 43x–1 = 1.5625 × 10–2.
6.In an arithmetic sequence, the first term is 5 and the fourth term is 40. Find the second term. 7.If loga 2 = x and loga 5 = y, find in terms of x and y, expressions for (a)log2 5;
8.Find the sum of the infinite geometric series
9.Find the coefficient of a5b7 in the expansion of (a + b)12. 10.The Acme insurance company sells two savings plans, Plan A and Plan B.
For Plan A, an investor starts with an initial deposit of $1000 and increases this by $80 each month, so that in the second month, the deposit is $1080, the next month it is $1160 and so on.
For Plan B, the investor again starts with $1000 and each month deposits 6% more than the previous month. (a)Write down the amount of money invested under Plan B in the second and third months.
Give your answers to parts (b) and (c) correct to the nearest dollar. (b)Find the amount of the 12th deposit for each Plan.
(c)Find the total amount of money invested during the first 12 months (i)under Plan A;
(ii)under Plan B.
11.$1000 is invested at the beginning of each year for 10 years.
The rate of interest is fixed at 7.5% per annum. Interest is compounded annually.
Calculate, giving your answers to the nearest dollar
(a)how much the first $1000 is worth at the end of the ten years; (b)the total value of the investments at the end of the ten years. 12.Let log10P = x , log10Q = y and log10R = z. Express in terms of x , y and z. 13.Each day a runner trains for a 10 km race. On the first day she runs 1000 m, and then increases the distance by 250 m on each subsequent day. (a)On which day does she run a distance of 10 km in training? (b)What is the total distance she will have run in training by the end of that day? Give your answer exactly. 14.Determine the constant term in the expansion of
15.Use the binomial theorem to complete this expansion.
(3x + 2y)4 = 81x4 + 216x3 y +...
16.The first three terms of an arithmetic sequence are 7, 9.5, 12. (a)What is the 41st term of the sequence?
(b)What is the sum of the first 101 terms of the sequence? 17.Solve the equation log9 81 + log9 + log9 3 = log9 x.
18.Consider the binomial expansion
(a)By substituting x = 1 into both sides, or otherwise, evaluate
19.Portable telephones are first sold in the country Cellmania in 1990. During 1990, the number of units sold is 160. In 1991, the number of units sold is 240 and in 1992, the number of units sold is 360.
In 1993 it was noticed that the annual sales formed a geometric sequence with first term 160, the 2nd and 3rd terms being 240 and 360 respectively. (a)What is the common ratio of this sequence?
Assume that this trend in sales continues.
(b)How many units will be sold during 2002?
(c)In what year does the number of units sold first exceed 5000? Between 1990 and 1992, the total number of units sold is 760. (d)What is the total number of units sold between 1990 and 2002?
During this period, the total population of Cellmania remains approximately 80 000. (e)Use this information to suggest a reason why the geometric growth in sales would not continue.
20.In an arithmetic sequence, the first term is –2, the fourth term is 16, and the nth term is 11 998. (a)Find the common difference d.
(b)Find the value of n.
21.Consider the expansion of
(a)How many terms are there in this expansion?
(b)Find the constant term in this expansion.
22.Solve the equation log27 x = 1 – log27 (x – 0.4).