Respuesta :
Answer:
First term = 1
Constant difference = 3
Step-by-step explanation:
To determine the first term and constant difference of an arithmetic series, we can begin by using the arithmetic series formula:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Sum of the first $n$ terms of an arithmetic series}}\\\\S_n=\dfrac{n}{2}[2a+(n-1)d]\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$a$ is the first term.}\\\phantom{ww}\bullet\;\textsf{$d$ is the common difference.}\\\phantom{ww}\bullet\;\textsf{$n$ is the position of the term.}\\\end{array}}[/tex]
Given that the sum of the first 10 terms is 145, we can substitute Sₙ = 145 and n = 10 into the sum formula to create an equation relating a to d:
[tex]\dfrac{10}{2}\left[2a+(10-1)d\right]=145[/tex]
Simplify:
[tex]5\left[2a+9d\right]=145[/tex]
[tex]2a+9d=29[/tex]
Now, we can use the general formula for the nth term of an arithmetic sequence to create expressions for the 3rd, 4th and 9th terms.
[tex]\boxed{\begin{array}{l}\underline{\textsf{General form of the $n$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]
Substitute n = 3, n = 4 and n = 9 into the formula:
[tex]a_3=a+(3-1)d=a+2d[/tex]
[tex]a_4=a+(4-1)d=a+3d[/tex]
[tex]a_9=a+(9-1)d=a+8d[/tex]
Given that the sum of the 4th and 9th terms is five times the 3rd term, then:
[tex](a+3d)+(a+8d)=5(a+2d)[/tex]
Simplify:
[tex]a+3d+a+8d=5a+10d[/tex]
[tex]2a+11d=5a+10d[/tex]
[tex]11d=3a+10d[/tex]
[tex]d=3a[/tex]
Now we have created a system of equations:
[tex]\begin{cases}2a+9d=29\\d=3a\end{cases}[/tex]
Substitute the second equation into the first equation, then solve for a:
[tex]2a+9(3a)=29[/tex]
[tex]2a+27a=29[/tex]
[tex]29a=29[/tex]
[tex]a=1[/tex]
Therefore, the first term is a = 1.
Substitute the found value of a into the second equation, and solve for d:
[tex]d=3(1)[/tex]
[tex]d=3[/tex]
Therefore, the common difference is d = 3.