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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
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