The <em>first</em> eight terms of the sequence are 27, 27, 39, 87, 117.378, 147.755, 178.132 and 208.509.
<h3>
Determination of a given set of successive values of a sequence</h3>
By (1) we have that
, and we simplify the system of equations as follows:
![(27-A)+d + A\cdot r = 27](https://tex.z-dn.net/?f=%2827-A%29%2Bd%20%2B%20A%5Ccdot%20r%20%3D%2027)
(2b)
(3b)
(4b)
By (2b), we simplify the system of equations once again:
(3c)
(4c)
And by equalising (3c) and (4c) we have an expression in terms of
:
![\frac{12}{[2\cdot (1-r)+(r^{2}-1)]} = \frac{60}{[3\cdot (1-r)+(r^{3}-1)]}](https://tex.z-dn.net/?f=%5Cfrac%7B12%7D%7B%5B2%5Ccdot%20%281-r%29%2B%28r%5E%7B2%7D-1%29%5D%7D%20%3D%20%5Cfrac%7B60%7D%7B%5B3%5Ccdot%20%281-r%29%2B%28r%5E%7B3%7D-1%29%5D%7D)
![12\cdot [3\cdot (1-r)+(r^{3}-1)] = 60\cdot [2\cdot (1-r) + (r^{2}-1)]](https://tex.z-dn.net/?f=12%5Ccdot%20%5B3%5Ccdot%20%281-r%29%2B%28r%5E%7B3%7D-1%29%5D%20%3D%2060%5Ccdot%20%5B2%5Ccdot%20%281-r%29%20%2B%20%28r%5E%7B2%7D-1%29%5D)
![36\cdot (1-r) +12\cdot (r^{3}-1) = 120\cdot (1-r)+60\cdot (r^{2}-1)](https://tex.z-dn.net/?f=36%5Ccdot%20%281-r%29%20%2B12%5Ccdot%20%28r%5E%7B3%7D-1%29%20%3D%20120%5Ccdot%20%281-r%29%2B60%5Ccdot%20%28r%5E%7B2%7D-1%29)
![84\cdot (1-r) +60\cdot (r^{2}-1)-12\cdot (r^{3}-1) = 0](https://tex.z-dn.net/?f=84%5Ccdot%20%281-r%29%20%2B60%5Ccdot%20%28r%5E%7B2%7D-1%29-12%5Ccdot%20%28r%5E%7B3%7D-1%29%20%3D%200)
![84-84\cdot r +60\cdot r^{2}-60-12\cdot r^{3}-12 = 0](https://tex.z-dn.net/?f=84-84%5Ccdot%20r%20%2B60%5Ccdot%20r%5E%7B2%7D-60-12%5Ccdot%20r%5E%7B3%7D-12%20%3D%200)
(5)
The roots of this <em>third order</em> polynomial are:
,
and
. Since
must be a <em>real</em> number, then
.
By (4c) we have the value of
:
![A = \frac{60}{3\cdot (1-0.0242)+(0.0242^{3}-1)}](https://tex.z-dn.net/?f=A%20%3D%20%5Cfrac%7B60%7D%7B3%5Ccdot%20%281-0.0242%29%2B%280.0242%5E%7B3%7D-1%29%7D)
![A \approx 31.130](https://tex.z-dn.net/?f=A%20%5Capprox%2031.130)
By (2b) we find the value of
:
![d = (1-0.0242)\cdot 31.130](https://tex.z-dn.net/?f=d%20%3D%20%281-0.0242%29%5Ccdot%2031.130)
![d = 30.377](https://tex.z-dn.net/?f=d%20%3D%2030.377)
And by (1) we find the value of
:
![a = 27-A](https://tex.z-dn.net/?f=a%20%3D%2027-A)
![a = 27-31.130](https://tex.z-dn.net/?f=a%20%3D%2027-31.130)
![a = -4.13](https://tex.z-dn.net/?f=a%20%3D%20-4.13)
The <em>first</em> eight terms are calculated below:
![n_{1} = 27](https://tex.z-dn.net/?f=n_%7B1%7D%20%3D%2027)
![n_{2} = 27](https://tex.z-dn.net/?f=n_%7B2%7D%20%3D%2027)
![n_{3} = 39](https://tex.z-dn.net/?f=n_%7B3%7D%20%3D%2039)
![n_{4} = 87](https://tex.z-dn.net/?f=n_%7B4%7D%20%3D%2087)
![n_{5} = [-4.13 + 4\cdot (30.377)]+(31.130)\cdot (0.0242)^{4} = 117.378](https://tex.z-dn.net/?f=n_%7B5%7D%20%3D%20%5B-4.13%20%2B%204%5Ccdot%20%2830.377%29%5D%2B%2831.130%29%5Ccdot%20%280.0242%29%5E%7B4%7D%20%3D%20117.378)
![n_{6} = [-4.13+5\cdot (30.377)+(31.130)\cdot (0.0242)^{5}] = 147.755](https://tex.z-dn.net/?f=n_%7B6%7D%20%3D%20%5B-4.13%2B5%5Ccdot%20%2830.377%29%2B%2831.130%29%5Ccdot%20%280.0242%29%5E%7B5%7D%5D%20%3D%20147.755)
![n_{7} = [-4.13+6\cdot (30.377)\cdot (31.130)\cdot (0.0242)^{6}] = 178.132](https://tex.z-dn.net/?f=n_%7B7%7D%20%3D%20%5B-4.13%2B6%5Ccdot%20%2830.377%29%5Ccdot%20%2831.130%29%5Ccdot%20%280.0242%29%5E%7B6%7D%5D%20%3D%20178.132)
![n_{8} = [-4.13+7\cdot (30.377)\cdot (31.130)\cdot (0.0242)^{7}] = 208.509](https://tex.z-dn.net/?f=n_%7B8%7D%20%3D%20%5B-4.13%2B7%5Ccdot%20%2830.377%29%5Ccdot%20%2831.130%29%5Ccdot%20%280.0242%29%5E%7B7%7D%5D%20%3D%20208.509)
The <em>first</em> eight terms of the sequence are 27, 27, 39, 87, 117.378, 147.755, 178.132 and 208.509. ![\blacksquare](https://tex.z-dn.net/?f=%5Cblacksquare)
<h3>
Remark</h3>
<em>The statement present typing mistakes and is poorly formatted. Correct form is shown below:</em>
<em />
<em>The first four terms of an arithmetic sequence are: </em>
<em>, </em>
<em>, </em>
<em>, </em>
<em>. The first four terms of another sequence are: </em>
<em>, </em>
<em>, </em>
<em>, </em>
<em>. The eight terms satisfy:</em>
<em />
<em />
<em> </em><em>(1)</em>
<em></em>
<em> </em><em>(2)</em>
<em></em>
<em> </em><em>(3)</em>
<em />
<em> </em><em>(4)</em>
<em></em>
<em>By using the substitution </em>
<em>, or otherwise, find the all eight terms. </em>
To learn more on sequences, we kindly invite to check this verified question: brainly.com/question/21961097