At Westonci.ca, we provide clear, reliable answers to all your questions. Join our vibrant community and get the solutions you need. Get quick and reliable solutions to your questions from a community of seasoned experts on our user-friendly platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
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
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.
