Question

Complete the arithmetic sequence and then find the nth term.
__,__,6,__,15,__, 24, ...​

Answer 1

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

`a_n=a+(n-1)d` ...(1)

Where, a is the first term and d is the common difference.

3rd term of the sequence is 6.

`a_3=a+(3-1)d`

`6=a+2d` ...(i)

5th term of the sequence is 15

`a_5=a+(5-1)d`

`15=a+4d` ...(ii)

Subtracting (i) from (ii), we get

`15-6=a+4d-a-2d`

`9=2d`

`(9)/(2)=d`

`4.5=d`

Putting d=4.5 in (i), we get

`6=a+2(4.5)`

`6=a+9`

`6-9=a`

`-3=a`

Putting a=-3 and d=4.5 in equation (1).

`a_n=-3+(n-1)4.5`

`a_n=-3+4.5n-4.5`

`a_n=4.5n-7.5`

The nth term of the given arithmetic sequence is
`a_n=4.5n-7.5`.

For n=1,

`a_1=4.5(1)-7.5`

`a_1=4.5-7.5`

`a_1=-3`

For n=2,

`a_2=4.5(2)-7.5`

`a_2=9-7.5`

`a_2=1.5`

For n=4,

`a_4=4.5(4)-7.5`

`a_4=18-7.5`

`a_4=10.5`

For n=6,

`a_6=4.5(6)-7.5`

`a_6=27-7.5`

`a_6=19.5`

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, ... .