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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Here is the explanation for the sequence and series question on Page 7: 1. If the first term of a G.P is 10 and the sum of the infinite terms is 20. Find the sum of the first 6 terms of the G.P. Understanding the problem: We are given the first term (a) and the sum to infinity (S_) of a Geometric Progression (GP). We need to find the sum of the first 6 terms (S_6). To do this, we first need to find the common ratio (r) using the sum to infinity formula, and then use the sum of the first n terms formula. Step 1: Write down the given information. The first term is a = 10. The sum of the infinite terms is S_ = 20. Step 2: Use the formula for the sum to infinity of a GP to find the common ratio (r). The formula for the sum to infinity of a GP is S_ = (a)/(1-r), provided that |r| < 1. Substitute the given values into the formula: 20 = (10)/(1-r) To solve for 1-r, multiply both sides by (1-r) and divide by 20: 1-r = (10)/(20) 1-r = (1)/(2) Now, solve for r: r = 1 - (1)/(2) r = (1)/(2) Since |r| = |(1)/(2)| < 1, this value of r is valid for an infinite sum. Step 3: Find the sum of the first 6 terms (S_6). The formula for the sum of the first n terms of a GP is S_n = (a(1 - r^n))/(1 - r). We use this form because |r| < 1. We have a=10, r=(1)/(2), and we want to find S_6, so n=6. Substitute these values into the formula: S_6 = (10 (1 - (1)/(2))^6)1 - (1)/(2) First, calculate ((1)/(2))^6: ((1)/(2))^6 = (1^6)/(2^6) = (1)/(64) Next, calculate the denominator 1 - (1)/(2): 1 - (1)/(2) = (1)/(2) Now substitute these back into the S_6 formula: S_6 = (10 (1 - 1)/(64))(1)/(2) Calculate the term in the parenthesis: 1 - (1)/(64) = (64)/(64) - (1)/(64) = (63)/(64) Substitute this back: S_6 = (10 (63)/(64))(1)/(2) To simplify, we can multiply by the reciprocal of the denominator: S_6 = 10 × (63)/(64) × 2 S_6 = (10 × 63 × 2)/(64) S_6 = (1260)/(64) Simplify the fraction by dividing both numerator and denominator by their greatest common divisor, which is 4: S_6 = (1260 ÷ 4)/(64 ÷ 4) S_6 = (315)/(16) The sum of the first 6 terms is (315)/(16).