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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.
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## 1 Answer

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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.
by Diamond (43,672 points)
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