Okay, let's break down the geometric progression concepts from question 4. A geometric progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous one by a constant, non-zero number called the common ratio ($r$). Here are the key formulas used in question 4: 1. The $n$-th term ($T_n$): This formula helps you find any term in the sequence if you know the first term ($a$) and the common ratio ($r$). $$ T_n = ar^{n-1} $$ In question 4(i), we used $T_3 = ar^2 = \frac{2}{9}$ and $T_4 = ar^3 = \frac{2}{27}$ to find $a$ and $r$. 2. The sum of the first $n$ terms ($S_n$): This formula calculates the total sum of a specific number of terms in the sequence. $$ S_n = \frac{a(1-r^n)}{1-r} \quad \text{ (when } r \neq 1 \text{)} $$ In question 4(ii), we used this formula with $a=2$, $r=\frac{1}{3}$, and $n=5$ to find the sum of the first 5 terms. 3. The sum to infinity ($S_\infty$): This formula applies when the common ratio $r$ has an absolute value less than 1 ($|r| < 1$). It calculates the sum of all terms in an infinitely long geometric progression. $$ S_\infty = \frac{a}{1-r} \quad \text{ (when } |r| < 1 \text{)} $$ In question 4(iii), we used this formula with $a=2$ and $r=\frac{1}{3}$ to find the sum to infinity.