Question #375bd

2 Answers
Dec 1, 2017

(n+1)(n+2)

Explanation:

I was thinking like

0\times1, 1\times2, 2\times3, 3\times4, 4\times5, 5\times6, 6\times7,...

So the means we have n^(th) term being (n+1)(n+2)

Dec 24, 2017

u_n = n^2-n

Explanation:

if the answer includes a variable to the power of 2, it is most likely a quadratic sequence.

the general formula for a quadratic sequence is u_n = an^2+bn+c

0, 2, 6, 12, 20, 30, 42

2, 4, 6, 8, 10, 12 (d_1)

2, 2, 2, 2, 2 (d_2)

the second difference is constant.

in an^2+bn+c, a = d_2/2

a = 2/2 = 1

this means that u_n = n^2+bn+c

b and c can be found by comparing the sequence in question with the sequence u_n = n^2, since a is the same for both of them.

u_n = n^2+bn+c:
0, 2, 6, 12, 20

u_n = n^2:
1,4,9,16,25

then, subtract the numbers in u_n = n^2 from u_n = n^2+bn+c:

0-1 = -1
2-4 = -2
6-9= -3
12-16 = -4

subtracting n^2 from u_n = n^2+bn+c gives us bn+c, which is a linear sequence.

-1, -2, -3, -4...

the nth term here is -n.

this means that bn+c = -n

-n is the same as -n +0, which gives c as 0, and b as -1.

finally, a= 1, b=-1, c=0
an^2+bn+c = n^2-n

u_n = an^2+bn+c = n^2-n