The three numbers are 4 , 8 , 16
Step-by-step explanation:
Let us revise how to find the nth term of the geometric and the arithmetic progressions
- The nth term of the geometric sequence is
, where a is the first term and r is the common ratio between the consecutive terms
- The nth term of the arithmetic sequence is
, where a is the first term and d in the common difference between the consecutive terms
∵ Three numbers form a geometric progression
- Assume that the first number is a and the common ratio is r
and n = 1 , 2 , 3
∴ The three terms are a , ar and ar²
∵ The second term is increased by 2
∴ The three terms are a , ar + 2 , ar²
∵ The three terms formed an arithmetic progression
- The common difference between each two consecutive terms is d
∴ d = ar + 2 - a and d = ar² - (ar + 2)
- Equate the right hand sides of d
∴ ar + 2 - a = ar² - ar - 2
- Add 2 to both sides
∴ ar - a + 4 = ar² - ar
- Subtract ar from both sides
∴ -a + 4 = ar² - 2ar
- Add a to both sides
∴ 4 = ar² - 2ar + a
- Take a as a common factor in the right hand side
∴ 4 = a(r² - 2r + 1)
∵ r² - 2r + 1 = (r - 1)²
∴ 4 = a(r - 1)²
- Divide both sides by (r - 1)²
∴ ⇒ (1)
∵ The last term is increased by 9
∴ The three terms are a , ar + 2 , ar² + 9
∵ The three terms formed an geometric progression
- Find the common ratio between each two consecutive terms
∵ The common ratio =
∵ The common ratio =
- Equate the right hand sides of the common ratio
∴
- By using cross multiplication
∴ a(ar² + 9) = (ar + 2)²
- Simplify the two sides
∴ a²r² + 9a = a²r² + 4ar + 4
- Subtract a²r² from both sides
∴ 9a = 4ar + 4
- Subtract 4ar from both sides
∴ 9a - 4ar = 4
- Take a as a common factor from both sides
∴ a(9 - 4r) = 4
- Divide both sides by (9 - 4r)
∴ ⇒ (2)
Equate the right hand sides of (1) and (2)
∴ =
- By using cross multiplication
∴ 4(r - 1)² = 4(9 - 4r)
- Divide both sides by 4
∴ (r - 1)² = 9 - 4r
- Solve the bracket of the left hand side
∴ r² - 2r + 1 = 9 - 4r
- Add 4r to both sides
∴ r² + 2r + 1 = 9
- Subtract 9 from both sides
∴ r² + 2r - 8 = 0
- Factorize it into 2 factors
∴ (r - 2)(r + 4) = 0
- Equate each factor by 0
∵ r - 2 = 0
- Add 2 to both sides
∴ r = 2
∵ r + 4 = 0
- Subtract 4 from both sides
∴ r = -4 ⇒ rejected
Substitute the value of r in equation (2) to find a
∵
∴ a = 4
∵ The three numbers are a , ar , ar²
∵ a = 4 and r = 2
∴ The numbers are 4 , 4(2) , 4(2)²
∴ The numbers are 4 , 8 , 16
Lets check the answer
4 , 8 + 2 , 16 ⇒ 4 , 10 , 16 formed an arithmetic progression with common difference 6
4 , 10 , 16 + 9 ⇒ 4 , 10 , 25 formed a geometric progression with common ratio 2.5
The three numbers are 4 , 8 , 16
Learn more:
You can learn more about the progressions in brainly.com/question/1522572
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