Answer:
[tex]-\frac{17}{4}[/tex] is 27th term of the given sequence.
Step-by-step explanation:
Given sequence is [tex]2\frac{1}{4},2,1\frac{3}{4},......[/tex]
Here, first term of the sequence,
[tex]a_1=2\frac{1}{4}[/tex]
By subtracting first term from the second term of the sequence,
[tex]T_2-T_1=2-2\frac{1}{4}[/tex]
= [tex](2-2)-\frac{1}{4}[/tex]
= [tex]-\frac{1}{4}[/tex]
Similarly, difference in second and third term,
[tex]1\frac{3}{4}-2=(1-2)+\frac{3}{4}[/tex]
= [tex]-1+\frac{3}{4}[/tex]
= [tex]\frac{3-4}{4}[/tex]
= [tex]-\frac{1}{4}[/tex]
Therefore, there is a common difference (d) of [tex](-\frac{1}{4})[/tex].
Hence, the sequence is an Arithmetic sequence.
Explicit formula of an Arithmetic sequence is given by,
[tex]T_n=a_1+(n-1)d[/tex]
Here, n = number of term
If, [tex]T_n=-\frac{17}{4}[/tex]
By substituting these values in the formula,
[tex]-\frac{17}{4}=2\frac{1}{4}+(n-1)(-\frac{1}{4})[/tex]
[tex]-\frac{17}{4}-2\frac{1}{4}=(n-1)(-\frac{1}{4})[/tex]
[tex]-\frac{17}{4}-\frac{9}{4}=-(n-1)(\frac{1}{4})[/tex]
[tex]-(\frac{17+9}{4})=(n-1)(\frac{1}{4})[/tex]
[tex]\frac{26}{4}=\frac{1}{4}(n-1)[/tex]
n - 1 = 26
n = 27
Therefore, [tex]-\frac{17}{4}[/tex] is 27th term of the given sequence.
