Answers − Quadratic Sequences

1.   Work out the 11^th term of the sequence 6, 9, 14, 21, 30, ...

(a)	Work out the difference between each term

	N	1		2		3		4		5
	Nth term	6		9		14		21		30
	1st differ.			+3		+5		+7		+9
	2nd differ.					+2		+2		+2

(b)	Work out the formula

As the 2nd difference is the same the sequence is a quadratic, try n2

	N	1		2		3		4		5
	N2	1		4		9		16		25
	differ.	+5		+5		+5		+5		+5
	Nth term	6		9		14		21		30
	So the nth term = n2 + 5

(c)	Work out the 11th term:

2.   Work out the 15^th term of the sequence −3, 0, 5, 12, 21, ...

(a)	Work out the difference between each term

	N	1		2		3		4		5
	Nth term	−3		0		5		12		21
	1st differ.			+3		+5		+7		+9
	2nd differ.					+2		+2		+2

(b)	Work out the formula

As the 2nd difference is the same the sequence is a quadratic, try n2

	N	1		2		3		4		5
	N2	1		4		9		16		25
	differ.	−4		−4		−4		−4		−4
	Nth term	−3		0		5		12		21
	So the nth term = n2 − 4

(c)	Work out the 15th term:

3.   Work out the 17^th term of the sequence −5, 1, 11, 25, 43, ...

(a)	Work out the difference between each term

	N	1		2		3		4		5
	Nth term	−5		1		11		25		43
	1st differ.			+6		+10		+14		+18
	2nd differ.					+4		+4		+4

(b)	Work out the formula

As the 2nd difference is the same the sequence is a quadratic, try 2n2

	N	1		2		3		4		5
	2n2	2		8		18		32		50
	differ.	−7		−7		−7		−7		−7
	Nth term	−5		1		11		25		43
	So the nth term = 2n2 − 7

(c)	Work out the 17th term:

4.   Work out the 9^th term of the sequence 14, 20, 30, 44, 62, ...

(a)	Work out the difference between each term

	N	1		2		3		4		5
	Nth term	14		20		30		44		62
	1st differ.			+6		+10		+14		+18
	2nd differ.					+4		+4		+4

(b)	Work out the formula

As the 2nd difference is the same the sequence is a quadratic, try 2n2

	N	1		2		3		4		5
	2n2	2		8		18		32		50
	differ.	+12		+12		+12		+12		+12
	Nth term	14		20		30		44		62
	So the nth term = 2n2 + 12

(c)	Work out the 9th term:

5.   Work out the 20^th term of the sequence 10, 19, 34, 55, 82, ...

(a)	Work out the difference between each term

	N	1		2		3		4		5
	Nth term	10		19		34		55		82
	1st differ.			+9		+15		+21		+27
	2nd differ.					+6		+6		+6

(b)	Work out the formula

As the 2nd difference is the same the sequence is a quadratic, try 3n2

	N	1		2		3		4		5
	3n2	3		12		27		48		75
	differ.	+7		+7		+7		+7		+7
	Nth term	10		19		34		55		82
	So the nth term = 3n2 + 7

(c)	Work out the 20th term:

6.   Work out the 19^th term of the sequence −9, 0, 15, 36, 63, ...

(a)	Work out the difference between each term

	N	1		2		3		4		5
	Nth term	−9		0		15		36		63
	1st differ.			+9		+15		+21		+27
	2nd differ.					+6		+6		+6

(b)	Work out the formula

As the 2nd difference is the same the sequence is a quadratic, try 3n2

	N	1		2		3		4		5
	3n2	3		12		27		48		75
	differ.	−12		−12		−12		−12		−12
	Nth term	−9		0		15		36		63
	So the nth term = 3n2 − 12

(c)	Work out the 19th term:

7.   Work out the 21^st term of the sequence 14, 23, 38, 59, 86, ...

(a)	Work out the difference between each term

	N	1		2		3		4		5
	Nth term	14		23		38		59		86
	1st differ.			+9		+15		+21		+27
	2nd differ.					+6		+6		+6

(b)	Work out the formula

As the 2nd difference is the same the sequence is a quadratic, try 3n2

	N	1		2		3		4		5
	3n2	3		12		27		48		75
	differ.	+11		+11		+11		+11		+11
	Nth term	14		23		38		59		86
	So the nth term = 3n2 + 11

(c)	Work out the 21st term:

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