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

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

$9=2d$

$\dfrac{9}{2}=d$

$4.5=d$

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, ... .