Question
These are the first four diagrams of a sequence.
The diagrams are made from white dots and black dots.
(a) Complete the table for Diagram 5 and Diagram 6.
(b) Write an expression, in terms of n, for the number of white dots in Diagram n.
………………………………………….
(c) The expression for the total number of dots in Diagram n is \(\frac{1}{2}(3n^{2}-n).\)
(i) Find the total number of dots in Diagram 8.
………………………………………….
(ii) Find an expression for the number of black dots in Diagram n.
Give your answer in its simplest form.
………………………………………….
(d) T is the total number of dots used to make all of the first n diagrams.
\(T=an^{3}+bn^{2}\)
Find the value of a and the value of b.
You must show all your working.
a = …………………………………………
b = …………………………………………
Answer/Explanation
(a) 25 36
10 15
35 51
(b) \(n^{2}\)
(c)(i) 92
(ii)\(\frac{1}{2}(n^{2}-n)\)
(d)\(a=\frac{1}{2}\),\(b=\frac{1}{2}\)
Question
Marco is making patterns with grey and white circular mats.
The patterns form a sequence.
Marco makes a table to show some information about the patterns.
(a) Complete the table for Pattern 5.
(b) Find an expression, in terms of n, for the number of grey mats in Pattern n.
…………………………………………
(c) Marco makes a pattern with 24 grey mats.
Find the total number of mats in this pattern.
…………………………………………
(d) Marco needs a total of 6 mats to make the first pattern.
He needs a total of 16 mats to make the first two patterns.
He needs a total of \(\frac{1}{6}n^{3}+an^{2}+bn \)mats to make the first n patterns.
Find the value of a and the value of b.
a = ………………………………………..
b = ………………………………………..
Answer/Explanation
(a) 18
28
(b) 3n + 3
(c) 45
(d)\(a=\frac{3}{2}\)
\(b=\frac{13}{3}\)
Question
On Monday, Ankuri sent this text message to two friends.
Today is Day Number 1.
Tomorrow, please add 1 to the Day Number and send this text message to two friends.
All the friends who receive a text message follow the instructions.
(a) Complete the table.
(b) Write down an expression for the number of text messages sent on Day Number n.
(c) Ankuri thinks that, by the end of Day Number 3, the total number of text messages that have been sent
is \(2^4 – 2\).
(i) Show that she is correct.
(ii) Complete the statement.
The total number of text messages sent by the end of Day Number 5 is …………………… which is
equal to \(2^k – 2\) where k = ………………………..
(iii) Write down an expression for the total number of text messages sent by the end of
Day Number n.
(iv) Find the Day Number when the total number of text messages sent by the end of the day is 1022.
Answer/Explanation
Answer:
(a) 4 5 6 7
8 16 32 64 128
(b) \(2^n\) oe
(c) (i) 2 + 4 + 8 = 14
16 – 2 = 14
(ii) 62
and
6
(iii) \(2^{n+1}-2\) oe
(iv) 9