Given:
The arithmetic sequence is
__, __, 6, __, 15, __, 24, ...
To find:
The missing values and the nth term of the sequence.
Solution:
The nth term of an arithmetic sequence is
[tex]a_n=a+(n-1)d[/tex] ...(1)
Where, a is the first term and d is the common difference.
3rd term of the sequence is 6.
[tex]a_3=a+(3-1)d[/tex]
[tex]6=a+2d[/tex] ...(i)
5th term of the sequence is 15
[tex]a_5=a+(5-1)d[/tex]
[tex]15=a+4d[/tex] ...(ii)
Subtracting (i) from (ii), we get
[tex]15-6=a+4d-a-2d[/tex]
[tex]9=2d[/tex]
[tex]\dfrac{9}{2}=d[/tex]
[tex]4.5=d[/tex]
Putting d=4.5 in (i), we get
[tex]6=a+2(4.5)[/tex]
[tex]6=a+9[/tex]
[tex]6-9=a[/tex]
[tex]-3=a[/tex]
Putting a=-3 and d=4.5 in equation (1).
[tex]a_n=-3+(n-1)4.5[/tex]
[tex]a_n=-3+4.5n-4.5[/tex]
[tex]a_n=4.5n-7.5[/tex]
The nth term of the given arithmetic sequence is [tex]a_n=4.5n-7.5[/tex].
For n=1,
[tex]a_1=4.5(1)-7.5[/tex]
[tex]a_1=4.5-7.5[/tex]
[tex]a_1=-3[/tex]
For n=2,
[tex]a_2=4.5(2)-7.5[/tex]
[tex]a_2=9-7.5[/tex]
[tex]a_2=1.5[/tex]
For n=4,
[tex]a_4=4.5(4)-7.5[/tex]
[tex]a_4=18-7.5[/tex]
[tex]a_4=10.5[/tex]
For n=6,
[tex]a_6=4.5(6)-7.5[/tex]
[tex]a_6=27-7.5[/tex]
[tex]a_6=19.5[/tex]
The missing values are -3, 1.5, 10.5, 19.5 respectively. So, the complete arithmetic sequence is -3, 1.5, 6, 10.5, 15, 19.5, 24, ... .
