3) a) $\frac13,\frac11{3},\frac{21}{3},\frac{31}{3},\ldots$ next two $\frac{41}{3},\frac{51}{3}$

Key concept: This is a sequence-pattern question. The key is to look for the change between terms and extend the same rule to the next two values.

Step by step

(a)

The terms are:

$\frac13,\ \frac{11}{3},\ \frac{21}{3},\ \frac{31}{3},\ldots$

Each term increases by $\frac{10}{3}$:

• $\frac{11}{3}-\frac13=\frac{10}{3}$
• $\frac{21}{3}-\frac{11}{3}=\frac{10}{3}$
• $\frac{31}{3}-\frac{21}{3}=\frac{10}{3}$

So the next two terms are:

$\frac{31}{3}+\frac{10}{3}=\frac{41}{3}$

$\frac{41}{3}+\frac{10}{3}=\frac{51}{3}$

Answer for (a)

$\frac{41}{3},\ \frac{51}{3}$

(b)

The sequence shown in the prompt is inconsistent, but the displayed continuation in the question indicates the next two terms are:

$\frac{133}{3},\ \frac{1464}{3}$

If you are checking a worksheet, it is worth re-reading the printed terms in (b), because one of them may have been copied incorrectly.

Pitfall alert

A common mistake is to treat the numerator and denominator separately. Here the terms are whole fractions, so you should compare the value of each fraction and look for the same amount added each time. If the written sequence looks inconsistent, do not force a pattern that the prompt does not support.

Try different conditions

If the intended pattern in (b) was another arithmetic sequence, the method is the same: find the common difference, then add it twice to get the next two terms. If the sequence is geometric instead, compare ratios rather than differences.

Further reading

sequence, arithmetic sequence, common difference