Welcome to Westonci.ca, where finding answers to your questions is made simple by our community of experts. Get accurate and detailed answers to your questions from a dedicated community of experts on our Q&A platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
The first eight terms of the sequence are 27, 27, 39, 87, 117.378, 147.755, 178.132 and 208.509.
Determination of a given set of successive values of a sequence
By (1) we have that [tex]a = 27 - A[/tex], and we simplify the system of equations as follows:
[tex](27-A)+d + A\cdot r = 27[/tex]
[tex]d + (r-1)\cdot A = 0[/tex] (2b)
[tex](27-A) +2\cdot d + A\cdot r^{2} = 39[/tex]
[tex]2\cdot d + (r^{2}-1)\cdot A = 12[/tex] (3b)
[tex](27-A) +3\cdot d + A\cdot r^{3} = 87[/tex]
[tex]3\cdot d + (r^{3}-1)\cdot A = 60[/tex] (4b)
By (2b), we simplify the system of equations once again:
[tex]2\cdot (1-r)\cdot A+(r^{2}-1)\cdot A = 12[/tex]
[tex][2\cdot (1-r)+(r^{2}-1)]\cdot A = 12[/tex] (3c)
[tex]3\cdot (1-r)\cdot A + (r^{3}-1)\cdot A = 60[/tex]
[tex][3\cdot (1-r) + (r^{3}-1)]\cdot A = 60[/tex] (4c)
And by equalising (3c) and (4c) we have an expression in terms of [tex]r[/tex]:
[tex]\frac{12}{[2\cdot (1-r)+(r^{2}-1)]} = \frac{60}{[3\cdot (1-r)+(r^{3}-1)]}[/tex]
[tex]12\cdot [3\cdot (1-r)+(r^{3}-1)] = 60\cdot [2\cdot (1-r) + (r^{2}-1)][/tex]
[tex]36\cdot (1-r) +12\cdot (r^{3}-1) = 120\cdot (1-r)+60\cdot (r^{2}-1)[/tex]
[tex]84\cdot (1-r) +60\cdot (r^{2}-1)-12\cdot (r^{3}-1) = 0[/tex]
[tex]84-84\cdot r +60\cdot r^{2}-60-12\cdot r^{3}-12 = 0[/tex]
[tex]-12\cdot r^{3}+60\cdot r^{2}-84\cdot r +2 = 0[/tex] (5)
The roots of this third order polynomial are: [tex]r_{1} \approx 0.0242[/tex], [tex]r_{2} = 2.488 + i\,0.831[/tex] and [tex]r_{3} \approx 2.488-i\,0.831[/tex]. Since [tex]r[/tex] must be a real number, then [tex]r \approx 0.0242[/tex].
By (4c) we have the value of [tex]A[/tex]:
[tex]A = \frac{60}{3\cdot (1-0.0242)+(0.0242^{3}-1)}[/tex]
[tex]A \approx 31.130[/tex]
By (2b) we find the value of [tex]d[/tex]:
[tex]d = (1-0.0242)\cdot 31.130[/tex]
[tex]d = 30.377[/tex]
And by (1) we find the value of [tex]a[/tex]:
[tex]a = 27-A[/tex]
[tex]a = 27-31.130[/tex]
[tex]a = -4.13[/tex]
The first eight terms are calculated below:
[tex]n_{1} = 27[/tex]
[tex]n_{2} = 27[/tex]
[tex]n_{3} = 39[/tex]
[tex]n_{4} = 87[/tex]
[tex]n_{5} = [-4.13 + 4\cdot (30.377)]+(31.130)\cdot (0.0242)^{4} = 117.378[/tex]
[tex]n_{6} = [-4.13+5\cdot (30.377)+(31.130)\cdot (0.0242)^{5}] = 147.755[/tex]
[tex]n_{7} = [-4.13+6\cdot (30.377)\cdot (31.130)\cdot (0.0242)^{6}] = 178.132[/tex]
[tex]n_{8} = [-4.13+7\cdot (30.377)\cdot (31.130)\cdot (0.0242)^{7}] = 208.509[/tex]
The first eight terms of the sequence are 27, 27, 39, 87, 117.378, 147.755, 178.132 and 208.509. [tex]\blacksquare[/tex]
Remark
The statement present typing mistakes and is poorly formatted. Correct form is shown below:
The first four terms of an arithmetic sequence are: [tex]a[/tex], [tex]a + d[/tex], [tex]a + 2\cdot d[/tex], [tex]a + 3d[/tex]. The first four terms of another sequence are: [tex]A[/tex], [tex]A\cdot r[/tex], [tex]A\cdot r^{2}[/tex], [tex]A\cdot r^{3}[/tex]. The eight terms satisfy:
[tex]a + A = 27[/tex] (1)
[tex](a+d)+A\cdot r = 27[/tex] (2)
[tex](a + 2\cdot d) + A\cdot r^{2} = 39[/tex] (3)
[tex](a + 3\cdot d) + A\cdot r^{3} = 87[/tex] (4)
By using the substitution [tex]a = 27-A[/tex], or otherwise, find the all eight terms.
To learn more on sequences, we kindly invite to check this verified question: https://brainly.com/question/21961097
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
