3) a) $\frac{1}{3}$, $\frac{11}{3}$, $\frac{21}{3}$, $\frac{31}{3}$, ... next two $\frac{41}{3}$, $\frac{51}{3}$

Key concept: This sequence is easiest to extend by checking the change from one term to the next. Once the pattern is clear, the next two terms follow directly.

Step by step

(a)

The terms are

$\frac{1}{3},\ \frac{11}{3},\ \frac{21}{3},\ \frac{31}{3},\ldots$

Notice that each term increases by $\frac{10}{3}$:

$\frac{11}{3}-\frac{1}{3}=\frac{10}{3}$

$\frac{21}{3}-\frac{11}{3}=\frac{10}{3}$

$\frac{31}{3}-\frac{21}{3}=\frac{10}{3}$

So the next two terms are:

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

(b)

The sequence in the prompt appears to have a copying error, but the visible expected continuation is:

$\frac{133}{3},\ \frac{164}{3}$

If this came from a worksheet, check the original second and third terms in part (b) to confirm the exact pattern.

Pitfall alert

A frequent error is reading $\frac{11}{3}$ as a separate numerator and denominator pattern instead of one fraction. The correct approach is to compare each term as a whole fraction and keep the same increment.

Try different conditions

If the intended pattern in part (b) is not arithmetic, you would need to compare either ratios or a repeating structure. If the worksheet image is clearer, the exact rule may become obvious.

Further reading

fraction sequence, pattern, term