Here are the first five terms of a sequence - Edexcel - GCSE Maths - Question 22 - 2017 - Paper 2

The general form of a quadratic expression can be expressed as:

$a_n = an^2 + bn + c$

Given the second difference of 4, we can deduce that:

a = \frac{4}{2} = 2.

Now we have:

a_n = 2n^2 + bn + c

Next, we will substitute the known values for n and a_n:

• For n = 1, a_1 = 4:
  $2(1)^2 + b(1) + c = 4$
  Thus,
  $2 + b + c = 4$
  ⇒
  $b + c = 2$ (Equation 1)

• For n = 2, a_2 = 11:
  $2(2)^2 + b(2) + c = 11$
  Thus, $8 + 2b + c = 11$ ⇒
  $2b + c = 3$ (Equation 2)

Now we can solve these equations simultaneously.

Using Equation 1:
$c = 2 - b$
Substituting into Equation 2: $2b + (2 - b) = 3$
⇒
$b + 2 = 3$
⇒
$b = 1$
Substituting back to find c: $c = 2 - 1 = 1$

Thus, we find: $a_n = 2n^2 + 1n + 1$.

The expression for the nth term of the sequence is:

$a_n = 2n^2 + n + 1$.