This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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4) The sum of the first n terms of a series is S_n = 5(2^n - 1). Find the general term (a_n) and show that the sequence is a G.P. Step 1: Find the general term a_n. The general term a_n is given by a_n = S_n - S_n-1 for n > 1, and a_1 = S_1. First, find a_1: S_1 = 5(2^1 - 1) = 5(2-1) = 5(1) = 5 So, a_1 = 5. Next, find S_n-1: S_n-1 = 5(2^n-1 - 1) Now, find a_n for n > 1: a_n = S_n - S_n-1 a_n = 5(2^n - 1) - 5(2^n-1 - 1) a_n = 5 [ (2^n - 1) - (2^n-1 - 1) ] a_n = 5 [ 2^n - 1 - 2^n-1 + 1 ] a_n = 5 [ 2^n - 2^n-1 ] a_n = 5 [ 2 · 2^n-1 - 2^n-1 ] a_n = 5 [ (2-1) · 2^n-1 ] a_n = 5 · 2^n-1 This formula also holds for n=1 (a_1 = 5 · 2^0 = 5). The general term is a_n = 5 · 2^n-1. Step 2: Show that the sequence is a Geometric Progression (G.P.). A sequence is a G.P. if the ratio of consecutive terms is constant. This constant ratio is the common ratio, r. We need to show that (a_n)/(a_n-1) is a constant for all n > 1. Using the general term a_n = 5 · 2^n-1: a_n-1 = 5 · 2^(n-1)-1 = 5 · 2^n-2 Now, calculate the ratio: (a_n)/(a_n-1) = 5 · 2^n-15 · 2^n-2 = 2^n-12^n-2 = 2^(n-1) - (n-2) = 2^n-1-n+2 = 2^1 = 2 Since the ratio of consecutive terms is a constant value of 2, the sequence is a Geometric Progression.