CH-09-001 · Quadratic sequence and membership test

The first four terms of a quadratic sequence are $21,\;51,\;95,\;153$ . Find an expression for the nth term. Determine whether 1305 is a term of the sequence, showing algebraic working.

Sequences: Quadratic nth Term Table & Membership Test Booklet p. 12, p. 25 & p. 26

Method

1) Use first and second differences

2) Subtract an² in a table to find the linear remainder

3) Equate the nth-term rule to the target

4) Accept only a positive integer position n

Worked Example (CH-09-001)

First differences:

30,\ 44,\ 58

. Second difference:

14

;

a=14\div2=7

n	1	2	3	4
n²	1	4	9	16
Sequence	21	51	95	153
7n²	7	28	63	112
Subtract	14	23	32	41

The remainder is linear with difference 9 and term before the first 5. Rule:

7n^2 + 9n + 5

Test membership:

7n^2 + 9n + 5=1305\Rightarrow 7n^2 + 9n - 1300=0

Multiply

a\times c=7\times(-1300)=-9100

. Choose the signed factor pair

-91,\ 100

: product

-9100

, sum

9

×	$n$	$-13$
$7n$	$7n^2$	$-91n$
$100$	$100n$	$-1300$

Read the factors outside the grid:

(7n + 100)(n - 13)

7n + 100=0\quad\Rightarrow\quad n=-\frac{100}{7}

n - 13=0\quad\Rightarrow\quad n=13

10)

n=13

is a positive integer, so this is term 13.

Ans =

7n^{2} + 9n + 5

; yes, term 13.