The 1^st and 6^th terms of a G.P are 153 and 17/27. Find the sum of the first four terms.

Here are detailed explanations for questions 6 and 7. 6. The 1^st and 6^th terms of a G.P are 153 and 17/27. Find the sum of the first four terms. Understanding the problem: We are given the first term (a) and the sixth term (T_6) of a Geometric Progression (GP). Our goal is to find the sum of the first four terms (S_4). To do this, we first need to find the common ratio (r) of the GP. Step 1: Write down the given information using the general term formula for a GP, T_n = a · r^n-1. The first term is T_1 = a. We are given T_1 = 153. So, a = 153. The sixth term is T_6 = a · r^6-1 = a · r^5. We are given T_6 = (17)/(27). So, we have the equation: a · r^5 = (17)/(27) Step 2: Find the common ratio (r). Substitute the value of a=153 into the equation from Step 1: 153 · r^5 = (17)/(27) To isolate r^5, divide both sides by 153: r^5 = (17)/(27 × 153) We can simplify the fraction. Notice that 153 = 9 × 17. r^5 = (17)/(27 × (9 × 17)) Cancel out the 17 from the numerator and denominator: r^5 = (1)/(27 × 9) Express the denominators as powers of 3: 27 = 3^3 and 9 = 3^2. r^5 = (1)/(3^3 × 3^2) Combine the powers of 3 in the denominator: r^5 = (1)/(3^3+2) r^5 = (1)/(3^5) Since r^5 = ((1)/(3))^5, we can conclude that: r = (1)/(3) Step 3: Find the sum of the first four terms (S_4). 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)/(3)| < 1. We have a=153, r=(1)/(3), and we want to find S_4, so n=4. Substitute these values into the formula: S_4 = (153 (1 - (1)/(3))^4)1 - (1)/(3) First, calculate ((1)/(3))^4: ((1)/(3))^4 = (1^4)/(3^4) = (1)/(81) Next, calculate the denominator 1 - (1)/(3): 1 - (1)/(3) = (3)/(3) - (1)/(3) = (2)/(3) Now substitute these back into the S_4 formula: S_4 = (153 (1 - 1)/(81))(2)/(3) Calculate the term in the parenthesis: 1 - (1)/(81) = (81)/(81) - (1)/(81) = (80)/(81) Substitute this back: S_4 = (153 (80)/(81))(2)/(3) To simplify, we can multiply by the reciprocal of the denominator: S_4 = 153 × (80)/(81) × (3)/(2) Now, perform the multiplication and simplification: S_4 = (153 × 80 × 3)/(81 × 2) We can simplify by dividing common factors. For example, 81 = 3 × 27, and 153 = 3 × 51 = 9 × 17. S_4 = ((9 × 17) × (2 × 40) × 3)/((9 × 9) × 2) Cancel out common factors (9 and 2): S_4 = (17 × 40 × 3)/(9) Further simplify by dividing 3 from the numerator and 9 from the denominator: S_4 = (17 × 40 × 1)/(3) S_4 = (680)/(3) The sum of the first four terms is (680)/(3). 7. If T_2=35 and T_4=875 respectively. Find the first and fifth terms. Understanding the problem: We are given the second term (T_2) and the fourth term (T_4) of a Geometric Progression (GP). We need to find the first term (a) and the fifth term (T_5). We will use the general term formula T_n = a · r^n-1 to set up equations and solve for a and r. Step 1: Write down the given information using the general term formula. For the second term (n=2): T_2 = a · r^2-1 = a · r = 35 (Equation 1) For the fourth term (n=4): T_4 = a · r^4-1 = a · r^3 = 875 (Equation 2) Step 2: Find the common ratio (r). To find r, we can divide Equation 2 by Equation 1. This eliminates a: (a · r^3)/(a · r) = (875)/(35) Simplify both sides: r^3-1 = (875)/(35) r^2 = 25 To solve for r, take the square root of both sides. Remember that a square root can be positive or negative: r = ± sqrt(25) r = ± 5 This means there are two possible values for the common ratio, r=5 and r=-5. We need to find the first and fifth terms for each case. Step 3: Find the first term (a) and the fifth term (T_5) for each possible value of r. Case 1: When r=5 Substitute r=5 into Equation 1 (a · r = 35): a · 5 = 35 Divide by 5 to find a: a = (35)/(5) a = 7 So, the first term is T_1 = a = 7. Now, find the fifth term (T_5) using T_n = a · r^n-1 with a=7, r=5, and n=5: T_5 = a · r^5-1 = a · r^4 T_5 = 7 · (5)^4 Calculate 5^4: 5 × 5 × 5 × 5 = 25 × 25 = 625. T_5 = 7 · 625 T_5 = 4375 Case 2: When r=-5 Substitute r=-5 into Equation 1 (a · r = 35): a · (-5) = 35 Divide by -5 to find a: a = (35)/(-5) a = -7 So, the first term is T_1 = a = -7. Now, find the fifth term (T_5) using T_n = a · r^n-1 with a=-7, r=-5, and n=5: T_5 = a · r^5-1 = a · r^4 T_5 = -7 · (-5)^4 Calculate (-5)^4: (-5) × (-5) × (-5) × (-5) = 25 × 25 = 625. (An even power of a negative number is positive). T_5 = -7 · 625 T_5 = -4375 Therefore, there are two possible solutions: If r=5, the first term is 7 and the fifth term is 4375. If r=-5, the first term is -7 and the fifth term is -4375.