Explore Westonci.ca, the premier Q&A site that helps you find precise answers to your questions, no matter the topic. Join our Q&A platform to connect with experts dedicated to providing precise answers to your questions in different areas. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
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 trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
