Question

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

Answer 1

let f(x) = ax^2 + bx + c

given:
f(2) = 4a + 2b + c = 1
f(3) = 9a + 3b + c = -6
f(5) = 25a + 5b + c = -14
subtract the 1st from the 2nd : 5a + b = -7
subtract the 2nd from the 3rd: 16a + 2b = -8 ===> 8a + b = -4

subtract these last two:
3a = 3 ====> a = 1
sub into 5a + b = -7
5 + b = -7 =====> b = -12

sub into the first:
4 - 24 + c = 1 ===> c = 21

f(x) = y = x^2 - 12x + 21

table of values:
x y ∆y ∆(∆y)
0 21
1 10 -11
2 1 -9 <b>2</b>
3 -6 -7 <b>2</b>
4 -11 -5 <b>2</b>
5 -14 -3 <b>2</b> ====> the 2nd difference is constant at 2

My mind locked in on this way of doing it, there probably is
an easier way

Answer 2

Hence, or otherwise, calculate the first term of the pattern