# Practice Math

Topics: Geometric progression, Natural number, Sequence Pages: 12 (1582 words) Published: April 21, 2013
1.

The nth term of an arithmetic sequence is given by un = 5 + 2n. (a) Write down the common difference. (1)

(b)

(i) (ii)

Given that the nth term of this sequence is 115, find the value of n. For this value of n, find the sum of the sequence. (5) (Total 6 marks)

2.

A sum of \$ 5000 is invested at a compound interest rate of 6.3 % per annum. (a) Write down an expression for the value of the investment after n full years. (1)

(b)

What will be the value of the investment at the end of five years? (1)

(c)

The value of the investment will exceed \$ 10 000 after n full years. (i) (ii) Write down an inequality to represent this information. Calculate the minimum value of n. (4) (Total 6 marks)

3.

(a)

Consider the geometric sequence −3, 6, −12, 24, …. (i) (ii) Write down the common ratio. Find the 15th term. (3)

Consider the sequence x − 3, x +1, 2x + 8, ….

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(b)

When x = 5, the sequence is geometric. (i) (ii) Write down the first three terms. Find the common ratio. (2)

(c)

Find the other value of x for which the sequence is geometric. (4)

(d)

For this value of x, find (i) (ii) the common ratio; the sum of the infinite sequence. (3) (Total 12 marks)

4.

Clara organizes cans in triangular piles, where each row has one less can than the row below. For example, the pile of 15 cans shown has 5 cans in the bottom row and 4 cans in the row above it.

(a)

A pile has 20 cans in the bottom row. Show that the pile contains 210 cans. (4)

(b)

There are 3240 cans in a pile. How many cans are in the bottom row? (4)

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(c)

(i)

There are S cans and they are organized in a triangular pile with n cans in the bottom row. Show that n2 + n − 2S = 0. Clara has 2100 cans. Explain why she cannot organize them in a triangular pile. (6) (Total 14 marks)

(ii)

5.

Ashley and Billie are swimmers training for a competition. (a) Ashley trains for 12 hours in the first week. She decides to increase the amount of time she spends training by 2 hours each week. Find the total number of hours she spends training during the first 15 weeks. (3)

(b)

Billie also trains for 12 hours in the first week. She decides to train for 10% longer each week than the previous week. (i) (ii) Show that in the third week she trains for 14.52 hours. Find the total number of hours she spends training during the first 15 weeks. (4)

(c)

In which week will the time Billie spends training first exceed 50 hours? (4) (Total 11 marks)

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6.

The diagram shows a square ABCD of side 4 cm. The midpoints P, Q, R, S of the sides are joined to form a second square.

A

Q

B

P

R

D
(a) (i) (ii) Show that PQ = 2 2 cm. Find the area of PQRS.

S

C

(3)

The midpoints W, X, Y, Z of the sides of PQRS are now joined to form a third square as shown. A W Q X B

P Y S

R

Z D

C

(b)

(i) (ii)

Write down the area of the third square, WXYZ. Show that the areas of ABCD, PQRS, and WXYZ form a geometric sequence. Find the common ratio of this sequence. (3)

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The process of forming smaller and smaller squares (by joining the midpoints) is continued indefinitely. (c) (i) (ii) Find the area of the 11th square. Calculate the sum of the areas of all the squares. (4) (Total 10 marks)

7.

Let f(x) = log3 (a)

x + log3 16 – log3 4, for x > 0. 2

Show that f(x) = log3 2x.
(2)

(b)

Find the value of f(0.5) and of f(4.5).
(3)

The function f can also be written in the form f(x) = (c) (i) Write down the value of a and of b.

ln ax . ln b

(ii)

Hence on graph paper, sketch the graph of f, for –5 ≤ x ≤ 5, –5 ≤ y ≤ 5, using a scale of 1 cm to 1 unit on each axis.

(iii)

Write down the equation of the asymptote.
(6)

(d)

Write down the value of f–1(0).
(1)

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The point A lies on the graph...