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Question

These diagrams represent the first three trapezium numbers (each diagram always starts with 2 dots in the top row)

1st
OO

2nd
OO
OOO
OOOO
3rd
OO
OOO
OOOO
OOOOO

The 3rd trapezium number is 9 because 9 dots can be arranged as a trapezium.
There are 2 dots in the top row, 3 dots in the next row and 4 dots in the bottom row.

a)Write down two other trapezium numbers.
b)i) Find an expression for the number of dots in the bottom row of the nth trapezium number.

ii) Test your expression for a suitable value of n.
iii) Find, with an explanation, an expression for the number of dots in the bottom of the diagram for the (n+1) th trapezium number.

The nth trapezium number is x.

c)i) Write down an expression in terms of n and x for the (n+1)th trapezium number.

ii) Test your expression for a suitable value of n.

d)Obtain, with an explanation or justification a general expression in terms of n for the nth trapezium number.

Answer 1

I think you left out:
OO
OOO

a)
The sequence should be:
2, 5, 9, 14, 20, 27, etc.

b) The number of dots in the bottom row:
1st number = 2
2nd number = 3
3rd number = 4
4th number = 5
etc.
i) So for n, there are n+1 dots in the bottom row.
ii) See above
iii) For n+1, there would be n+2 dots in the bottom row. Just substitute (n+1) in place of n. So you have (n+1) + 1 = n+2.

c) The nth trapezium number is x
i) The (n+1)th trapezium number is x + (n+2)
ii) Let's try the 3rd trapezium number (9). n=3, x=9.
The 4th trapezium number should be x + n + 2 = 9 + 3 + 2 = 14.
This checks with our list before.

d) Notice how the trapezium number is the average of the top and bottom row, times the number of rows.
1st number: 1 row x (2 + 2)/2 = 2
2nd number: 2 rows x (2 + 3)/2 = 5
3rd number: 3 rows x (2 + 4)/2 = 9

So in general:
nth trapezium number: n rows x (2 + n+1)/2

Simplifying:
n(n+3)/2

Double checking our formula:
n = 1 --> 1(1+3)/2 = 4/2 = 2
n = 2 --> 2(2+3)/2 = 10/2 = 5
n = 3 --> 3(3+3)/2 = 18/2 = 9
n = 4 --> 4(4+3)/2 = 28/2 = 14
n = 5 --> 5(5+3)/2 = 40/2 = 20
n = 6 --> 6(6+3)/2 = 54/2 = 27
etc.

Answer 2

2, 5, 9, 14, ... (In fact, you missed 5, 2+3, which should be the second T(n=2))

a. 20, 27 for the fifth and sixth T

b. n + 1; n = 1....N; the bases are a simple arithmetic progression because you add one dot to the previous row on each successive T; e.g., the base for T(n = 5) = 6. QED.

c. Total dots (x) in the nth T is x(n = 1) = 2; x(2) = x(1) + 3; x(3) = x(2) + 4; x(4) = x(3) + 5...x(n+1) = x(n) + (n + 1): 9 = 5 + 4 QED

d. x(4) = x(3) + 5 = x(2) + 4 + 5 = x(1) + 3 + 4 + 5 = 2 + 3 + 4 + 5; so that x(N) = sum(n + 1); n = 1 through N + 1; e.g., x(5) = 2 + 3 + 4 + 5 + 6 = 20, as specified in a. QED

Answer 3

bottom row of nth trapezium number = n+1
T(n) = 2+3+4+...+(n+1) = (n/2)(n+3)

Answer 4

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Answer 5

That's not a challenge! That's a Brain twister!! Who put this kind of hell on you my friend? There really mean whoever it is. I'll see if I can get one of my Friends over here...likes puzzles and things of the sort....

Answer 6

Here is the first part for you
n(x) = .5x^2 + 1.5x

Answer 7

Too tough for me. Sorry!! I tried, but my brain is frozen now.

Good luck

Answer 8

well, i don't think i can be doing that right now. i'm busy with rocket science at the moment

Answer 9

tol!!! masyadong mahaba yan!!! that's too long!!!