The 29th term of an AP is -5, find the common difference if its first term is 31.

The formula for the $n$-th term ($T_n$) of an AP is:

$$T_n = a + (n-1)d$$

where $a$ is the first term, $n$ is the term number, and $d$ is the common difference.

We are given:

• The 28th term ($T_{28}$) is -5. So, $T_n = -5$ and $n = 28$.
• The first term ($a$) is 31. So, $a = 31$. We need to find the common difference ($d$).

$$-5 = 31 + (27)d$$ $$-5 = 31 + 27d$$

Subtract 31 from both sides:

$$-5 - 31 = 27d$$ $$-36 = 27d$$

Divide by 27:

$$d = \frac{-36}{27}$$

Simplify the fraction by dividing both numerator and denominator by their greatest common divisor, which is 9:

$$d = \frac{-36 \div 9}{27 \div 9}$$ $$d = \frac{-4}{3}$$

The common difference is $-\frac{4}{3}$.