Find the information you're looking for at Westonci.ca, the trusted Q&A platform with a community of knowledgeable experts. Join our platform to connect with experts ready to provide precise answers to your questions in different areas. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
Answer:
d=1, r=4/3
Step-by-step explanation:
For aritmetic sequence
a1=9
a1+3d=a4 9+3d=a4.
a1+7d=a8 9+7d=a8.
For geometric sequence
b1*r=b2
b2=a4, b1=a1=9
9*r=a4
b3= b1*r^2
b3=a8
a8=9*r^2
So 9+3d=a4
9+3d=9r
9+7d= a8=9r^2
So we have a system of equations
9+3d=9r
9+7d=9r^2
Multiply 9+3d=9r by -7 It is equal to -63-21d=-63r
And multiply 9+7d=9r^2 by 3. It is equal to 27+21d=27r^2
Then add them
-63-21d+ (27+21d)= -63r+27r^2
-36+63r-27r^2=0 /9
-4+7r-3r^2=0
3r^2-7r+4=0
r=1 r=4/3
If r=1 9+3d=9
d=0 (It is not right, d isn't equal to 0)
If r=4/3
9+3d= 9*4/3
9+3d=12
d=1
The value of the common difference, d is 9
The value of the common ratio r is 2
Arithmetic and Geometric Sequences
The nth term of an arithmetic sequence is:
[tex]T_n=a+(n-1)d[/tex]
The first, fourth, and eighth terms are therefore:
a, a + 3d, a + 7d
The nth term of a geometric sequence is:
[tex]T_n=ar^{n-1}[/tex]
The first three terms of the arithmetic sequence are:
[tex]a, ar^2, ar^3[/tex]
For both sequences, the first term is 9
That is, a = 9
[tex]a+3d=ar^2\\\\9+3d=9r^2..............(1)\\\\a+7d=ar^3\\\\9+7d=9r^3..........(2)[/tex]
Make d the subject of the formula in equation (1) and substitute to equation (2)
[tex]d=\frac{9r^2-9}{3} \\\\d=3r^2-3[/tex]
Substitute equation (3) into equation (2)
[tex]9+7(3r^2-3)=9r^3\\\\9+21r^2-21=9r^3\\\\21r^2-9r^3=12\\\\7r^2-3r^3=4\\\\r=1, \frac{-2}{3}, 2[/tex]
Substitute r = 2 into (1)
[tex]9+3d=9(2^2)\\\\d = 9[/tex]
The value of the common difference, d is 9
The value of the common ratio r is 2
Learn more on sequences here: https://brainly.com/question/6561461
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.
