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

Answer:

The 105th term of given sequence is $\frac{105}{2}$.

Step-by-step explanation:

The given sequence is

$\frac{1}{2},1,\frac{3}{2},2$

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.

$a_n=0.5(105)$

$a_n=52.5$

$a_n=\frac{105}{2}$

Therefore the 105th term of given sequence is $\frac{105}{2}$.