Question

find the next term in this arithmetic sequence.
\frac{7}{2}, \frac{9}{4}, 1, -\frac{1}{4}, -\frac{3}{2}, -\frac{?}{}

Explanation:

Step1: Convert terms to 4 denominator

Rewrite all terms with denominator 4:
$\frac{7}{2} = \frac{14}{4}$, $\frac{9}{4} = \frac{9}{4}$, $1 = \frac{4}{4}$, $-\frac{1}{4} = -\frac{1}{4}$, $-\frac{3}{2} = -\frac{6}{4}$

Step2: Calculate common difference

Subtract consecutive terms:
$\frac{9}{4} - \frac{14}{4} = -\frac{5}{4}$
Verify: $\frac{4}{4} - \frac{9}{4} = -\frac{5}{4}$, $-\frac{1}{4} - \frac{4}{4} = -\frac{5}{4}$, $-\frac{6}{4} - (-\frac{1}{4}) = -\frac{5}{4}$

Step3: Find next term

Add common difference to last term:
$-\frac{6}{4} + (-\frac{5}{4}) = -\frac{11}{4}$

Answer:

$\frac{11}{4}$ (so the missing numerator is 11, denominator is 4)