Welcome to Westonci.ca, where you can find answers to all your questions from a community of experienced professionals. Join our platform to connect with experts ready to provide detailed answers to your questions in various areas. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.
Sagot :
The geometric and arithmetic progression are a number series such that
each term is given by adding or multiplying the previous term by a constant.
Responses:
The possible three numbers are;
- 4, 8, 16
- [tex]\displaystyle \frac{4}{25} , \ -\frac{16}{25}, \ \frac{64}{25}[/tex]
Methods by which the above responses are obtained:
Let the three numbers be a, a·r, and a·r², we have;
Given;
G.P. = a, a·r, a·r²
A.P. = a, a·r + 2, a·r²
G.P. = a, a·r + 2, a·r² + 9
Solution:
(a·r + 2) - a = a·r² - (a·r + 2) by definition of an AP
a·r + 2 - a = a·r² - (a·r + 2)
(r² - 2·r + 1)·a - 4 = 0
G.P = a, a·r + 2, a·r² + 9
[tex]\displaystyle \frac{a \cdot r + 2}{a} = \frac{a \cdot r^2 + 9}{a \cdot r + 2}[/tex]
(a·r + 2)² = a·(a·r² + 9)
r²·a² + 4·r·a + 4 = r²·a² + 9·a
4·r·a + 4 = 9·a
4 = 9·a - 4·r·a = a·(9 - 4·r)
[tex]\displaystyle a = \mathbf{ \frac{4}{9 - 4 \cdot r}}[/tex]
[tex]\displaystyle \left(r^2 - 2 \cdot r + 1 \right) \times \frac{4}{9 - 4 \cdot r} - 4 = 0[/tex]
Therefore;
[tex]\displaystyle \left(r^2 - 2 \cdot r + 1 \right) \times \frac{4}{9 - 4 \cdot r} \times (9 - 4 \cdot r) - 4 \times (9 - 4 \cdot r)= 0[/tex]
4·r² + 8·r - 32 = 0
Which gives;
4·r² + 8·r - 32 = 4·(r - 2)·(r + 4) = 0
r = 2 or r = -4
[tex]\displaystyle a = \frac{4}{9 - 4 \times 2} = \mathbf{ 4}[/tex] or [tex]\displaystyle a = \frac{4}{9 - 4 \times (-4)} = \frac{4}{25}[/tex]
The first three numbers are;
4, 4 × 2, 4 × 2², which gives;
- 4, 8, 16
[tex]\displaystyle When \ a = \frac{4}{25}, and \ r = -4, we \ have;[/tex]
[tex]\displaystyle \frac{4}{25}, \ \frac{4}{25} \times (-4), \ \frac{4}{25} \times (-4)^2[/tex]
Which gives;
[tex]\displaystyle \frac{4}{25} , \ -\frac{16}{25} , \ \frac{64}{25}[/tex]
[tex]\displaystyle The \ second \ three \ numbers \ are; \underline{\frac{4}{25}, \ -\frac{16}{25} , \ \frac{64}{25}}[/tex]
Learn more about geometric and arithmetic progression here:
https://brainly.com/question/1800090
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.
