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

105th term of given series is

`a_n=(105)/(2)`

Explanation:

Given series is

`(1)/(2),\ 1,\ (3)/(2),\ 2,\ (5)/(2).....`

As we can see,

`\textrm{First term},a_1=(1)/(2)`

Also,

`1-(1)/(2)=(3)/(2)-1=2-(3)/(3)=.....=(1)/(2)`

hence, we can say given series is in arithmetic progression,

with common difference,

`d=\ (1)/(2)`

As given in question the nth term in A.P is given by

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

since we have to find the 105th term, so we can write

`a_(105)=(1)/(2)+(105-1)(1)/(2)`

`=(1)/(2)+(104)/(2)`

`=(105)/(2)`

Hence, the 105th term of given series of A.P is
`(105)/(2)`.