Given the number pattern: 20; 18: 14; 8;
a) Determine the nth term of this number pattern.
b) Determine the value of T12 in this number pattern.
c) Which term in this number pattern will have a value of - 36?

A quadratic number pattern has a second term equal to 1, a third term equal to -6 and a fifth term equal to - 14.
a) Calculate the second difference of this quadratic number pattern.
b) Hence, or otherwise, calculate the first term of this number pattern.

You have two problems involving quadratic sequences. For the sequence that starts 20, 18, 14, 8, ..., you want the n-th term, the 12-th term, and the term number of -36. For the sequence with terms 2, 3, and 5 having values 1, -6, and -14, you want the second difference and the first term.

1. 20, 18, 14, 8

The first attachment shows the first and second differences of this sequence each begin with -2. The first of differences at level n can be put into a formula for the n-th term:

∆0 +(n -1)(∆1 +(n -2)/2(∆2 + ...))

We have (∆0, ∆1, ∆2) = (20, -2, -2), so the expression for the n-th term is ...

tn = 20 +(n -1)(-2 +(n -2)/2(-2))

(a) tn = -n² +n +20

Listing the first 12 terms, we find the 12th term is ...

(b) t12 = -112

Locating the term -36 in the list, we find ...

(c) -36 is term 8

2. x, 1, -6, y, -14

For a quadratic sequence the third differences are zero. The second attachment shows us the third differences for this sequence are ...

-3(y +11) = 0 ⇒ y = -11

y -x +21 = 0 ⇒ x = 10

That attachment also shows us the second differences are x-8 (or y+13):

x -8 = 10 -8 = 2

(a) The second difference is 2.

(b) The first term is 10.

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Additional comment

The ability of this free calculator app to perform arithmetic symbolically and to find differences of successive list elements is very helpful for solving questions related to sequences. The linear and quadratic regression capabilities can also be useful for some questions. It pays to know your tools.

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