Number Sequence - help me please?

Question

I was just going to finish off my revision when I came to a question, which I haven no idea how to tackle. It might be sooo very easy, i don't know.. i just have no idea what to do, please help, thank you! x


1) Work out the difference between the nth number and the (n+1)th number. Give your answer as simply as you can in terms of n.

The sequence: 2,6,12,20


(Note: In the previous question, it asked me to find the nth term of the sequence and that I found to be n^2 + n ...)

ok.. i found out that the answer is: 2n + 2 or 2(n+1) ... but i still don't get how to work it out.

Thanks for all your help - i'm trying to understand it.. hopefully it'll click into my head aaanny second now. =)

Answer 1

Looks to me like f(1) = 2, f(n+1) = f(n) + 2(n+1). This is the sequence in recursive notation. I chose to do it recursively so that we would have an explicit relationship between f(n+1) and f(n). Subtracting, we have f(n+1) - f(n) = 2(n+1). So 2(n+1) is the difference between the (n+1)th number and the (n)th number.

Answer 2

Just look at the difference between the numbers.
2->6 is 4
6->12 is 6
12->20 is 8

So since it's counting up with even numbers, 2*n is a good candidate. A good place to start would be seeing if you can add 2*n to the previous number to get the next answer (or n = (n-1) + 2*n)

Let's see if we can rewrite the sequence in those terms:
f(2) = f(n-1) + 2^n => 2 + 2*2 = 6
f(3) = f(n-1) + 2^n => 6 + 2*3 = 12

Looks like your answer is:

f(1) = 2
f(n) = f(n-1) + 2*n

Answer 3

OK
Term Value
1 2
2 6
3 12
4 20

I notice that the value = n*(n+1)

So the next term, the 5th term = 5 * 6 = 30.

Answer 4

The difference between first and second - 4, second and third - 6, 3 and 4 - 8, so the difference between n and n+1 will be 2*(n+1)

Answer 5

Un: 2, 6, 12, 20, ......
∆1: .. 4 , 6 , 8, ......
∆2: ...., 2 , 2, .....

Under the given information the diferences become constant at the second level so the general term is of degree 2

So Un = an² + bn + c

So U1 = a*1 + b*1 + c

ie 2 = a + b + c ....... Equation (1)

U2 = 4a + 2b + c

ie 6 = 4a + 2b + c ....... Equation (2)
12 = 9a + 3b + c ....... Equation (3)
20 = 16a + 4b + c ....... Equation (4)

So Equation (2) - Equation (1)

4 = 3a + b ....... Equation (5)

Equation (3) - Equation (2)

6 = 5a + b ....... Equation (6)

Equation (6) - Equation (5)

2 = 2a so a = 1
Substitute in Equation (5) 4 = 3 + b
so b = 1

Substitute both solutions in Equation (1)

2 = 2 + c so c = 0

So Un = n² + n = n(n+1)

Check U1 = 2
U4 = 20

So your previous solution was correct

Answer 6

Factoring,

2 = 1*2

6 = 2*3

12 = 3*4

20 = 4*5

next one looks like 5*6 = 30

Answer 7

I chosen 6 to be 1st area and 18 the 2d and befor 6 it relatively is going to likely be area 0 (0, and in the previous ther area -a million.... X (new area)=X (previous area) * (n+a million) new area is n previous area is n-a million as an occasion, X placed interior the 4th area is 360 it relatively is the recent area. We have been given the fee in this new area, n, via multipling the X in previous area, it relatively is seventy two via the recent area +a million, meanning n+a million=4+a million=5 seventy two*5=360 and 360*6=2160 and 2160*7=15,a hundred and twenty and.......... or, the fee interior the nth place is 6/2*(n+a million)!=3*(n+a million)! or Xn=3*(n+a million)! For X>=0 or n>=-a million So in area 5 that's 6*5*4*3*2 * 3 = 2,a hundred and sixty and in the previous 6 that's 3 and in the previous that that's 0