Question

A quadratic pattern has a second term equal to 1, a third term equal to -6 and a fifth term equal to -14

1. Calculate the second difference of this quadratic pattern
2. Hence, or otherwise, calculate the first term of the pattern.

Answer 1

1. Let $1^{\text{st}}$ and $4^{\text{th}}$ terms be $x$ and $y$ respectively,

$1^{\text{st}} \text{difference} = 1-x; -7; y+6; -14-y; ..$

$2^{\text{nd}} \text{difference} = x-8; y+13;  -2y-20; ...$

since the $2^{\text{nd}}$ difference is constant,

$x-8=y+13$   ... Equation 1

$y+13=-2y-20$  ... Equation 2

then from Equation 2,

\[3y=-33\]

            \[ y= -11\]

and substitute in Equation 1,   

\[x= -11+13+8= 10\]

hence the $2^{\text{nd}} \text{difference} = y+13= -11+13= 2$

2. $1^{\text{st}} \text{term} = x$

                 $= 10$ , calculated in question 1.