Everything Maths Grade 11

5.2 CHAPTER 5. QUADRATIC SEQUENCES

where a, b and c are some constants to be determined.

Tn = an^2 + bn + c

T 1 = a(1)^2 + b(1) + c

= a + b + c (5.3)

T 2 = a(2)^2 + b(2) + c

= 4a + 2b + c (5.4)

T 3 = a(3)^2 + b(3) + c

= 9a + 3b + c (5.5)

The first difference (d) is obtained from

Let d≡ T 2 − T 1

∴ d = 3a + b

⇒ b = d− 3 a (5.6)

The common second difference (D) is obtained from

D = (T 3 − T 2 )− (T 2 − T 1 )

= (5a + b)− (3a + b)

= 2a

⇒ a =

D

2

(5.7)

Therefore, from (5.6),

b = d−

3

2

. D (5.8)

From (5.3),

c = T 1 − (a + b) = T 1 −

D

2

− d +

3

2

. D

∴ c = T 1 + D− d (5.9)

Finally, the general equation for the nth-term of a quadratic sequence is given by

Tn=

D

2

. n^2 + (d−

3

2

D). n + (T 1 − d + D) (5.10)

Example 3: Using a set of equations

QUESTION

Study the following pattern: 1;7;19;37;61;...

1. What is the next number in the sequence?


2. Use variables to write an algebraic statement to generalise the pattern.


3. What will the 100 thterm of the sequence be?