yoda82
Answered

Discover the best answers at Westonci.ca, where experts share their insights and knowledge with you. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.

In a G.P the difference between the 1st and 5th term is 150, and the difference between the

2nd and the 4th terms is 48. Find the sum of the first five terms.
​

Sagot :

jacob193

Answer:

Either [tex]\displaystyle \frac{-1522}{\sqrt{41}}[/tex] (approximately [tex]-238[/tex]) or [tex]\displaystyle \frac{1522}{\sqrt{41}}[/tex] (approximately [tex]238[/tex].)

Step-by-step explanation:

Let [tex]a[/tex] denote the first term of this geometric series, and let [tex]r[/tex] denote the common ratio of this geometric series.

The first five terms of this series would be:

  • [tex]a[/tex],
  • [tex]a\cdot r[/tex],
  • [tex]a \cdot r^2[/tex],
  • [tex]a \cdot r^3[/tex],
  • [tex]a \cdot r^4[/tex].

First equation:

[tex]a\, r^4 - a = 150[/tex].

Second equation:

[tex]a\, r^3 - a\, r = 48[/tex].

Rewrite and simplify the first equation.

[tex]\begin{aligned}& a\, r^4 - a \\ &= a\, \left(r^4 - 1\right)\\ &= a\, \left(r^2 - 1\right) \, \left(r^2 + 1\right) \end{aligned}[/tex].

Therefore, the first equation becomes:

[tex]a\, \left(r^2 - 1\right) \, \left(r^2 + 1\right) = 150[/tex]..

Similarly, rewrite and simplify the second equation:

[tex]\begin{aligned}&a\, r^3 - a\, r\\ &= a\, \left( r^3 - r\right) \\ &= a\, r\, \left(r^2 - 1\right) \end{aligned}[/tex].

Therefore, the second equation becomes:

[tex]a\, r\, \left(r^2 - 1\right) = 48[/tex].

Take the quotient between these two equations:

[tex]\begin{aligned}\frac{a\, \left(r^2 - 1\right) \, \left(r^2 + 1\right)}{a\cdot r\, \left(r^2 - 1\right)} = \frac{150}{48}\end{aligned}[/tex].

Simplify and solve for [tex]r[/tex]:

[tex]\displaystyle \frac{r^2+ 1}{r} = \frac{25}{8}[/tex].

[tex]8\, r^2 - 25\, r + 8 = 0[/tex].

Either [tex]\displaystyle r = \frac{25 - 3\, \sqrt{41}}{16}[/tex] or [tex]\displaystyle r = \frac{25 + 3\, \sqrt{41}}{16}[/tex].

Assume that [tex]\displaystyle r = \frac{25 - 3\, \sqrt{41}}{16}[/tex]. Substitute back to either of the two original equations to show that [tex]\displaystyle a = -\frac{497\, \sqrt{41}}{41} - 75[/tex].

Calculate the sum of the first five terms:

[tex]\begin{aligned} &a + a\cdot r + a\cdot r^2 + a\cdot r^3 + a \cdot r^4\\ &= -\frac{1522\sqrt{41}}{41} \approx -238\end{aligned}[/tex].

Similarly, assume that [tex]\displaystyle r = \frac{25 + 3\, \sqrt{41}}{16}[/tex]. Substitute back to either of the two original equations to show that [tex]\displaystyle a = \frac{497\, \sqrt{41}}{41} - 75[/tex].

Calculate the sum of the first five terms:

[tex]\begin{aligned} &a + a\cdot r + a\cdot r^2 + a\cdot r^3 + a \cdot r^4\\ &= \frac{1522\sqrt{41}}{41} \approx 238\end{aligned}[/tex].

We hope this was helpful. Please come back whenever you need more information or answers to your queries. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.