Question

Find the nth term of each sequence:
a) 2, 6, 12, 20, ...
b) -5, -2, 3, 10, ...

Answer 1

Answer:

2 , 6 , 12 , 20...

n = 1

a1 = 2 = 1 + 1 = 1 ^ 2 + 1 = n ^ 2 + n

n = 2

a2 = 6 = 4 + 2 = 2 ^ 2 + 2 = n ^ 2 + n

n = 3

a3 = 12 = 9 + 3 = 3 ^ 2 + 3 = n ^ 2 + n

n = 4

a3 = 20 = 16 + 4 = 4 ^ 2 + 4 = n ^ 2 + n

an = n ^ 2 + n

Answer 2

Answer:

Given the sequence 2,6,12,20, write it out in a line with gaps between each term:

2     6     12      20

Add a line of term below listing the differences between each pair of terms:

2     6     12      20

    4      6       8

Add another line of term below listing the differences between each pair of terms:

2     6     12     20

     4     6      8

           2      2

Notice that both terms of the last line are the same.

That implies that our given sequence can be matched using a quadratic formula, which we can construct using the first term of each of the lines as a coefficient:

an=0!2+1!4(n−1)+2!2(n−1)(n−2)

=2+4n−4+n2−3n+2

=n2+n

If you know the formulas for n=1ΣNn2 and n=1ΣNn then you can just add them to provide the formula for n=1ΣNan

I would rather construct it directly:

Add another line to the top of our sequences, consisting of the sums of the terms on the original top line:

0     2     8     20  40

     2     6   12     20

       4      6      8

            2        2

We want to start with 

Step-by-step explanation:

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