Question

A quadratic pattern has a second term equal to 1,a third term equal to -16 and a fifth term equal to -14

1.1.1.calculate the second difference of this quadratic pattern

1.1.2.hence or otherwise,calculate the first term of term

Answer 1

1.1.1. The common difference is 12

1.1.2. The first term is 30

Explanation:

1.1.1. The given quadratic pattern is presented as follows;

The 2nd term = 1

The 3rd term = -16

The 5th term = -14

Let 'd' represent the second difference, we have;

1 + 2d + a = -16

2·d + a = -16 - 1 = -17

2·d + a = -17...(1)

-16 + 3d + a + 4d + a = -14

7·d + 2·a = -14 + 16 = 2

7·d + 2·a = 2...(2)

Making 'a' the subject of both equation (1) and (2) gives;

a = -17 - 2·d

a = 1 - 7/2·d

∴ -17 - 2·d = 1 - 7/2·d

(7/2)·d - 2·d = 1 + 17

(3/2)·d = 18

d = (2/3) × 18 = 12

The common difference, d = 12

a = -17 - 2·d = -17 - 2 × 12 = -41

1.1.2. Let 'x' represent the first term, we have;

x + a + d = 1

x = 1 - (a + d)

x = 1 - (-41 + 12) = 30

The first term, x = 30