Sequences and the nth Term: Question 3

Worked solution

Step 1: Confirm the sequence is linear

Find the differences between consecutive terms:

$9-2=7,\quad 16-9=7,\quad 23-16=7,\quad 30-23=7.$

The first difference is constant and equal to $7$, so this is a linear (arithmetic) sequence with common difference $d=7$. Its $n$th term therefore has the form:

$u_n = 7n + c,$

where $c$ is a constant we must determine.

Step 2: Find the constant $c$

Substitute a known term. Using the first term $u_1 = 2$ (so $n=1$):

$7(1) + c = 2 \;\Longrightarrow\; 7 + c = 2 \;\Longrightarrow\; c = 2 - 7 = -5.$

Hence:

$u_n = 7n - 5.$

Step 3: Verify

Check every listed term:

Every value matches.

Alternative method (zero term)

A quick shortcut is to find the “zero term”: the term that would sit before the first one. Going back one step from the first term subtracts the common difference:

$u_0 = 2 - 7 = -5.$

Then the $n$th term is

$u_n = (\text{common difference})\times n + (\text{zero term}) = 7n + (-5) = 7n - 5,$

which agrees with Step 2.

Why the other options fail

• $7n+5$: correct gradient, wrong sign on the constant; gives $12,19,26,\ldots$
• $7n+2$: uses the first term $2$ as the constant instead of the zero term $-5$; gives $9,16,23,\ldots$ (shifted one place).
• $5n-7$: swaps the common difference and the constant; gives $-2,3,8,\ldots$

Final answer

$\boxed{u_n = 7n - 5} \quad\text{(Option A)}$