Respuesta :
Answer: a₈=36
Step-by-step explanation:
[tex]\displaystyle\\\left \{ {{a_1+a_7=40} \atop {a_1*a_4=160}} \right. \\\\\\\left \{ {{a_1+a_1+6d=40} \atop {a_1*a_4=160}} \right.\\\\\\\left \{ {{2a_1+6d=40\ (divide \ both\ sides \ of \ the\ equation\ by \ 2)} \atop {a_1*a_4=160}} \right. \\\\\\\left \{ {{a_1+3d=20=a_4} \atop {a_1*20=160\ (divide \ both\ sides \ of \ the\ equation\ by \ 20)}} \right. \\\\\\\left \{ {{8+3d=20} \atop {a_1=8}} \right. \\\\\\[/tex]
[tex]\displaystyle\\\left \{ {{3d=12\ (divide \ both\ sides \ of \ the\ equation\ by \ 3)} \atop {a_1=8}} \\\right.\\\\\\\left \{ {{d=4} \atop {a_1=8}} \right.\ \ \ \ \ \ Hence,\\\\a_8=a_1+7d\\\\a_8=8+7*4\\\\a_8=8+28\\\\a_8=36[/tex]
Answer:
a₈ = 36
Step-by-step explanation:
The general formula for the nth term of an arithmetic sequence is:
[tex]\boxed{\begin{array}{l}\underline{\textsf{$n$th term of an arithmetic sequence}}\\\\a_n=a+(n-1)d\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$a_n$ is the nth term.}\\ \phantom{ww}\bullet\;\textsf{$a$ is the first term.}\\\phantom{ww}\bullet\;\textsf{$d$ is the common difference between terms.}\\\phantom{ww}\bullet\;\textsf{$n$ is the position of the term.}\\\end{array}}[/tex]
Begin by creating expressions for the first, fourth and seventh terms by substituting n = 1, n = 4 and n = 7 into the formula:
[tex]a_1=a[/tex]
[tex]a_4=a+3d[/tex]
[tex]a_7=a+6d[/tex]
Given that the sum of the first and seventh terms is 40, then:
[tex]a_1 + a_7 = 40 \\\\ a + a + 6d = 40 \\\\ 2a + 6d = 40 \\\\ 2(a + 3d) = 40 \\\\ a + 3d = 20[/tex]
Rearrange the equation to isolate 3d:
[tex]3d = 20 - a[/tex]
Given that the product of the first and fourth terms is 160, then:
[tex]a_1 \cdot a_4 = 160 \\\\ a(a + 3d) = 160 \\\\ a^2 + 3da = 160[/tex]
Substitute 3d = 20 - a and solve for a:
[tex]a^2 + (20 - a)a = 160 \\\\ a^2 + 20a - a^2 = 160 \\\\ 20a = 160 \\\\ a &= 8[/tex]
Substitute a = 8 into 3d = 20 - a and solve for d:
[tex]3d = 20 - 8 \\\\ 3d = 12 \\\\ d = 4[/tex]
Now we have found the initial term (a = 8) and the common difference (d = 4), we can plug these into the formula to create an equation for the nth term of the sequence:
[tex]a_n = 8 + (n - 1)(4)[/tex]
[tex]a_n = 8 + 4n - 4[/tex]
[tex]a_n = 4n +4[/tex]
Finally, substitute n = 8 into the equation to find the value of the 8th term:
[tex]a_8 = 4(8) +4[/tex]
[tex]a_8 = 32 +4[/tex]
[tex]a_8 = 36[/tex]
Therefore, the value of the 8th term is:
[tex]\Large\boxed{\boxed{a_8 = 36}}[/tex]
