Answer:
Common difference: d = y + 1
Explicit form: aₙ = n(y + 1) + 2
9th term: a₉ = 9y + 11
Step-by-step explanation:
The explicit form of an arithmetic sequence allows us to find any term in the sequence without having to know the preceding terms. The general formula is:
[tex]\boxed{\begin{array}{l}\underline{\textsf{General formula for 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]
The first five terms of an arithmetic sequence are given as:
[tex]a_1=y+3\\\\a_2=2y+4\\\\a_3=3y+5\\\\a_4=4y+6\\\\a_5=5y+7[/tex]
To find the common difference (d), subtract any term from the term that follows it. Let's subtract the first term from the second term:
[tex]d = a_2-a_1\\\\d=(2y+4)-(y+3)\\\\d=2y+4-y-3\\\\d=y+1[/tex]
Therefore, the common difference is:
[tex]\Large\boxed{\boxed{d = y + 1}}[/tex]
To write the explicit form, substitute a = y + 3 and d = y + 1 into the general formula:
[tex]a_n=(y+3)+(n-1)(y+1)[/tex]
Simplify:
[tex]a_n=y+3+ny+n-y-1\\\\a_n=2+ny+n\\\\a_n=n(y+1)+2[/tex]
Therefore, the simplified explicit form is:
[tex]\Large\boxed{\boxed{a_n= n(y + 1) + 2}}[/tex]
To find the 9th term, substitute n = 9 into the nth term equation:
[tex]a_9=9(y+1)+2\\\\a_9=9y+9+2\\\\a_9=9y+11[/tex]
Therefore, the 9th term of the given arithmetic sequence is:
[tex]\Large\boxed{\boxed{a_9=9y+11}}[/tex]
