Quadratic sequence help

[Troy_12]

How would I create a quadratic sequence for each of these different questions? Could someone please explain it to me thanks!!! They're both separate

1) The first term is 3

2)Every term is a negative number

 

[tkhunny]

1) Pick a starting value - We are given 3

Sequence so far: 3

2) Pick a first, first difference. No, I'm not stuttering. How about 2.

Sequence so far: 3, 3+2 ==> 3, 5

3) Pick a second difference. How about 1.

Now iterate first differences

Second first difference is 2 + 1 = 3
Third first difference is 3 + 1 = 4
Fourth first difference is 4 + 1 = 5

Now iterate for final values

3
5
5+3 = 8
8+4 = 12
12+5 = 17

Sequence so far: 3, 5, 8, 12, 17

Check first differnces

5-3 = 2
8-5 = 3
12-8 = 4
17-12 = 5

Check second differences

3-2 = 1
4-3 = 1
5-4 = 1

It's looking pretty good.

 

[mmm4444bot]

Are you familiar with taking second differences between terms of a quadratic sequence and discovering that they're equal?

That process can be reversed.

In other words, you could start with some arbitrary constant, and work backwards to create pairs of numbers that differ by that amount. When you create the next set of pairs, just start with 3.

If you're not familiar with taking second differences, then you won't know what I'm talking about. Here's another way.

Look at the standard form of a quadratic equation:

y = Ax^2 + Bx + C

If we restrict the domain of x to the set of Natural numbers, then the range of y is a quadratic sequence. It's easy to pick values for the parameters A, B, and C, such that y = 3 when x = 1.

Once you have a sequence for exercise (1), can you simply multiply each term by -1, to obtain a sequence for exercise (2) ?