An exponential sequence GP has a positive common ratio. If the sum of the infinity sequence is 25 and the sum of the first two terms is 16, find the (a) 5th term (b) The sum of the first four terms.

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Asked on June 5, 2026Mathematics

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.

Accepted Answer · 2026-06-05T06:08:47.633Z

Step 1: Set up equations for the given information. The sum to infinity of a GP is given by S_ = (a)/(1-r), where a is the first term and r is the common ratio. The sum of the first n terms of a GP is given by S_n = (a(1-r^n))/(1-r). We are given: S_ = 25 (a)/(1-r) = 25 (*) The sum of the first two terms is S_2 = a + ar = a(1+r). We are given: S_2 = 16 a(1+r) = 16 (**) Step 2: Solve for a and r. From equation (*), we can express a in terms of r: a = 25(1-r) Substitute this expression for a into equation (**): 25(1-r)(1+r) = 16 25(1-r^2) = 16 1-r^2 = (16)/(25) r^2 = 1 - (16)/(25) r^2 = (25-16)/(25) r^2 = (9)/(25) r = ±sqrt((9)/(25)) r = ±(3)/(5) Since the problem states that the common ratio is positive, we take r = (3)/(5). Now substitute r = (3)/(5) back into the expression for a: a = 25(1-(3)/(5)) a = 25((5-3)/(5)) a = 25((2)/(5)) a = 5 × 2 a = 10 So, the first term is a=10 and the common ratio is r=(3)/(5). a) The 5th term Step 3: Calculate the 5th term. The formula for the n-th term of a GP is T_n = ar^n-1. For the 5th term (n=5): T_5 = ar^5-1 = ar^4 T_5 = 10 ((3)/(5))^4 T_5 = 10 × (3^4)/(5^4) T_5 = 10 × (81)/(625) T_5 = (810)/(625) Simplify the fraction by dividing the numerator and denominator by 5: T_5 = (162)/(125) The 5th term is (162)/(125). b) The sum of the first four terms Step 4: Calculate the sum of the first four terms. The formula for the sum of the first n terms of a GP is S_n = (a(1-r^n))/(1-r). For the sum of the first four terms (n=4): S_4 = (10(1-(3)/(5))^4)1-(3)/(5) S_4 = (10(1-81)/(625))(2)/(5) S_4 = (10(625-81)/(625))(2)/(5) S_4 = (10(544)/(625))(2)/(5) S_4 = 10 × (544)/(625) × (5)/(2) S_4 = (10 × 544 × 5)/(625 × 2) S_4 = (50 × 544)/(1250) S_4 = (27200)/(1250) Simplify the fraction by dividing the numerator and denominator by 50: S_4 = (544)/(25) The sum of the first four terms is (544)/(25). Send me the next one 📸

Accepted Answer · 2026-06-05T06:08:47.633Z

Step 1: Set up equations for the given information. The sum to infinity of a GP is given by S_ = (a)/(1-r), where a is the first term and r is the common ratio. The sum of the first n terms of a GP is given by S_n = (a(1-r^n))/(1-r). We are given: S_ = 25 (a)/(1-r) = 25 (*) The sum of the first two terms is S_2 = a + ar = a(1+r). We are given: S_2 = 16 a(1+r) = 16 (**) Step 2: Solve for a and r. From equation (*), we can express a in terms of r: a = 25(1-r) Substitute this expression for a into equation (**): 25(1-r)(1+r) = 16 25(1-r^2) = 16 1-r^2 = (16)/(25) r^2 = 1 - (16)/(25) r^2 = (25-16)/(25) r^2 = (9)/(25) r = ±sqrt((9)/(25)) r = ±(3)/(5) Since the problem states that the common ratio is positive, we take r = (3)/(5). Now substitute r = (3)/(5) back into the expression for a: a = 25(1-(3)/(5)) a = 25((5-3)/(5)) a = 25((2)/(5)) a = 5 × 2 a = 10 So, the first term is a=10 and the common ratio is r=(3)/(5). a) The 5th term Step 3: Calculate the 5th term. The formula for the n-th term of a GP is T_n = ar^n-1. For the 5th term (n=5): T_5 = ar^5-1 = ar^4 T_5 = 10 ((3)/(5))^4 T_5 = 10 × (3^4)/(5^4) T_5 = 10 × (81)/(625) T_5 = (810)/(625) Simplify the fraction by dividing the numerator and denominator by 5: T_5 = (162)/(125) The 5th term is (162)/(125). b) The sum of the first four terms Step 4: Calculate the sum of the first four terms. The formula for the sum of the first n terms of a GP is S_n = (a(1-r^n))/(1-r). For the sum of the first four terms (n=4): S_4 = (10(1-(3)/(5))^4)1-(3)/(5) S_4 = (10(1-81)/(625))(2)/(5) S_4 = (10(625-81)/(625))(2)/(5) S_4 = (10(544)/(625))(2)/(5) S_4 = 10 × (544)/(625) × (5)/(2) S_4 = (10 × 544 × 5)/(625 × 2) S_4 = (50 × 544)/(1250) S_4 = (27200)/(1250) Simplify the fraction by dividing the numerator and denominator by 50: S_4 = (544)/(25) The sum of the first four terms is (544)/(25). Send me the next one 📸

An exponential sequence GP has a positive common ratio. If the sum of the infinity sequence is 25 and the sum of the first two terms is 16, find the (a) 5th term (b) The sum of the first four terms.

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An exponential sequence GP has a positive common ratio. If the sum of the infinity sequence is 25 and the sum of the first two terms is 16, find the (a) 5th term (b) The sum of the first four terms.

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