Sequences and the nth Term: Question 4

Worked solution

Part (a): find the $n$th term

Step 1: Identify the common difference.

$51 - 58 = -7, \qquad 44 - 51 = -7, \qquad 37 - 44 = -7$

The terms decrease by the same amount each time, so this is a linear sequence with common difference $d = -7$.

Step 2: Start with the multiplier.

Since the common difference is $-7$, the $n$th term contains $-7n$. Compare $-7n$ with the actual sequence:

Step 3: Find the constant to add.

Each actual term is $65$ more than $-7n$ (since $58-(-7)=65$). So we add $65$:

$n\text{th term} = -7n + 65$

Quick check: for $n=4$, $-7(4)+65 = -28+65 = 37$ ✓

$\boxed{n\text{th term} = -7n + 65}$

Part (b): find $n$ when the term is $-19$

Set the expression from part (a) equal to $-19$:

$-7n + 65 = -19$

Subtract $65$ from both sides:

$-7n = -19 - 65 = -84$

Divide both sides by $-7$:

$n = \frac{-84}{-7} = 12$

Check: $-7(12) + 65 = -84 + 65 = -19$ ✓

$\boxed{n = 12}$

Final answers

• (a) $n$th term $= -7n + 65$
• (b) $n = 12$