Westonci.ca offers fast, accurate answers to your questions. Join our community and get the insights you need now. Experience the convenience of getting reliable answers to your questions from a vast network of knowledgeable experts. Get detailed and accurate answers to your questions from a dedicated 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
We hope this was helpful. Please come back whenever you need more information or answers to your queries. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.
