Question

the sequence 4;9;x;37;...is a quadratic sequence. a) calculate x. b) determine nth term of the sequence

Answer 1

4, 9, x, 37
first difference:
5, x-9, 37-x
2nd difference
x-14, 37-x - (x-9)
x-14, 46-2x

To be a quadratic sequence, the 2nd differences must all be the same, so

x-14 = 46-2x
3x = 60
x = 20

so our terms are 4, 9, 20, 37

Answer 2

general term for the above:
let term(n) = an^2 + bn + c
for n = 1
4 = a+b+c **
for n = 2
9 = 4a + 2b + c ***
for n = 3
20 = 9a + 3b + c ****

subtract ** from ***
5 = 3a + b ##

subtract *** from ****
11 = 5a + b ###

subtract ## from ###
6 = 2a
a = 3

into ##
5 = 9+b
b = -4

into **
4 = 3 - 4 + c
c = 5

term(n) = 3n^2 - 4n + 5

check for term(4)
LS = 37
RS = 3(16) - 16 + 5 = 37
my answer is correct