Fill in the blank part 2
Answer no. 3
Topic: Arithmetic Sequence
If it is not visible in the pic the nth term for no.3 is 9​

Fill in the blank part 2 Answer no 3 Topic Arithmetic Sequence If it is not visible in the pic the nth term for no3 is 9 class=


Answer :

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]