Question

In an arithmetic​ sequence, the nth term an is given by the formula an=a1+(n−1)d​, where a1 is the first term and d is the common difference.​ Similarly, in a geometric​ sequence, the nth term is given by 1an=a1•rn−1​,

where r is the common ratio. Use these formulas to determine

the indicated term in the given sequence.

The 105th term of 1/2, 1, 3/2, 2,..

Answer 1

The 105th term of given sequence is
`(105)/(2)`.

Explanation:

The given sequence is

`(1)/(2),1,(3)/(2),2`

It can be rewritten as

`0.5,1,1.5,2`

Here the first term is 0.5.

It is an arithmetic​ sequence because it has common difference.

`d=a_2-a_1=1-0.5=0.5`

`d=a_3-a_2=1.5-1=0.5`

`d=a_4-a_3=2-1.5=0.5`

The nth term of an AP is

`a_n=a_1+(n-1)d`

where,
`a_1` is first term and d is common difference.

Substitute
`a_1=0.5` and
`d=0.5` in the above formula.

`a_n=0.5+(n-1)0.5`

`a_n=0.5+0.5n-0.5`

`a_n=0.5n`

We need to find the 105th term of given sequence.

Substitute n=105 in the above equation.

`a_n=0.5(105)`

`a_n=52.5`

`a_n=(105)/(2)`

Therefore the 105th term of given sequence is
`(105)/(2)`.