Sequences and the nth Term: Question 2

Worked solution

Part (a): Number of tiles in Pattern 5

Write out the sequence and find the first differences (gaps between consecutive terms):

$6 \;\xrightarrow{+7}\; 13 \;\xrightarrow{+11}\; 24 \;\xrightarrow{+15}\; 39$

The first differences are $7,\ 11,\ 15$. These are not constant, so the sequence is not linear. Now find the second differences (gaps between the first differences):

$7 \;\xrightarrow{+4}\; 11 \;\xrightarrow{+4}\; 15$

The second differences are constant at $4$, which confirms the sequence is quadratic.

To get the next term, continue the pattern of differences. The next first difference is

$15 + 4 = 19,$

so the number of tiles in Pattern 5 is

$39 + 19 = \boxed{58}.$

Part (b): nth term of the sequence

Step 1: Find the coefficient of $n^2$. For a quadratic sequence $an^2 + bn + c$, the constant second difference equals $2a$. Here the second difference is $4$, so

$2a = 4 \quad\Longrightarrow\quad a = 2.$

Step 2: Remove the $2n^2$ part. Subtract $2n^2$ from each term to find what is left:

Step 3: Identify the linear remainder. The values $4,5,6,7$ form a linear (arithmetic) sequence going up by $1$ each time. Its $n$th term is

$1\cdot n + 3 = n + 3.$

Step 4: Combine.

$\text{nth term} = 2n^2 + (n + 3) = 2n^2 + n + 3.$

Check: at $n = 1$, $2(1)^2 + 1 + 3 = 6$ ✓; at $n = 4$, $2(16) + 4 + 3 = 39$ ✓; at $n = 5$, $2(25) + 5 + 3 = 58$ ✓ (agrees with part (a)).

$\boxed{2n^2 + n + 3}$

Alternative method (simultaneous equations): Substitute $n=1,2,3$ into $an^2+bn+c$: $a+b+c=6,\quad 4a+2b+c=13,\quad 9a+3b+c=24.$ Subtracting in pairs gives $3a+b=7$ and $5a+b=11$, so $2a=4\Rightarrow a=2$, then $b=1$ and $c=3$, giving the same result.

Part (c): Which pattern uses 303 tiles?

Set the $n$th term equal to $303$:

$2n^2 + n + 3 = 303.$

Rearrange into standard quadratic form:

$2n^2 + n - 300 = 0.$

Solve by factorising. We need two numbers multiplying to $2 \times (-300) = -600$ and adding to $1$; these are $25$ and $-24$:

$2n^2 + 25n - 24n - 300 = 0$ $n(2n + 25) - 12(2n + 25) = 0$ $(2n + 25)(n - 12) = 0.$

So

$n = -\tfrac{25}{2} \quad\text{or}\quad n = 12.$

Since $n$ must be a positive whole number (it is a pattern number), we reject $n = -\tfrac{25}{2}$.

(Check via the quadratic formula: $n = \dfrac{-1 \pm \sqrt{1^2 - 4(2)(-300)}}{2(2)} = \dfrac{-1 \pm \sqrt{2401}}{4} = \dfrac{-1 \pm 49}{4}$, giving $n = 12$ or $n = -12.5$.)

Verification: $2(12)^2 + 12 + 3 = 288 + 12 + 3 = 303$ ✓.

$\boxed{n = 12 \;\text{(Pattern 12)}}$