Question

Here are the first six terms of a quadratic sequence

10 19 34 55 82 115

Find an expression, in terms of n, for the nth term of this sequence.

Answer 1

nth term = 3n² + 7

Explanation:

• We can represent the nth term of a quadratic sequence as
  an² + bn + c where a, b and c are constants

• Plug in the given values for the first second and third terms (n = 1, 2 and 3) and solve for a, b and c to get the general expression for the nth term

• For n = 1,
  an² + bn + c = a(1²) + (1)b + c
  => a + b + c = 10 [1]

• For n = 2,
  an² + bn + c = a(2²) + (2)b + c = 19
  => 4a + 2b + c = 19 [2]

• Subtract [1] from [2] to get
  4a + 2b + c - (a + b + c) = 19 - 10
  => 4a + 2b + c - a - b - c = 9
  => 4a - a + 2b - b + c - c = 9
  => 3a + b = 9 [3]

• For n = 3,
  an² + bn + c = a(3²) + (3)b + c
  => 9a + 3b + c = 34

• 9a + 3b = 3(3a + b) by factoring out 3

• So
  9a + 3b + c = 34 becomes
  3(3a + b) + c = 34

• From eq 3 we have 3a + b = 9
  Substituting for 3a + b we get
  3(9) + c = 34
  => 27 + c = 24
  => c = 34 - 27= 7

• 
  Plug this value of c into equation [1], a + b + c = 10 to get
  => a + b + 7 = 10
  => a + b = 10 - 7
  => a + b = 3 [4]

• We have equations 3 and 4 as
  3a + b = 9 [3]
  a + b = 3 [4]

• Subtract equation [4] from [3] to get
  [3] - [4]
  => 3a + b - (a + b) = 9 - 3
  => 2a + 0 = 6
  => a = 6/2 = 3

• Plug the values of a = 3 and c = 7 into equation 1 to solve for b
  a + b + c = 10
  3 + b + 7 = 10
  10 + b = 10
  b = 0

• So the quadratic sequence nth term is given by the expression
  3n² + 0n + 7
  
  which simplifies to
  3n² + 7