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

Step 1: Recall the formula for the $n$-th term of an Arithmetic Progression (AP). 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.

Step 2: Identify the given values from the problem. 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 ($d$).

Step 3: Substitute the given values into the formula. $-5 = 31 + (28-1)d$

Step 4: Simplify the equation. $-5 = 31 + (27)d$ $-5 = 31 + 27d$

Step 5: Solve for $d$. 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 $-\frac{4}{3}$.

$\boxed{-\frac{4}{3}}$ That's 2 down. 3 left today — send the next one.