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).

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 = (2)/(9) and T_4 = ar^3 = (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 = (a(1-r^n))/(1-r) (when r ≠ 1 ) In question 4(ii), we used this formula with a=2, r=(1)/(3), and n=5 to find the sum of the first 5 terms. 3. The sum to infinity (S_): 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_ = (a)/(1-r) (when |r| < 1 ) In question 4(iii), we used this formula with a=2 and r=(1)/(3) to find the sum to infinity.