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 determinethe indicated term in the given sequence.The 105th term of 1/2, 1, 3/2, 2,..

Accepted Solution

Answer:105th term of given series is[tex]a_n=\dfrac{105}{2}[/tex]Step-by-step explanation:Given series is[tex]\dfrac{1}{2},\ 1,\ \dfrac{3}{2},\ 2,\ \dfrac{5}{2}.....[/tex]As we can see,[tex]\textrm{First term},a_1=\dfrac{1}{2}[/tex]Also,[tex]1-\dfrac{1}{2}=\dfrac{3}{2}-1=2-\dfrac{3}{3}=.....=\dfrac{1}{2}[/tex]hence, we can say given series is in arithmetic progression,with common difference,[tex]d=\ \dfrac{1}{2}[/tex]As given in question the nth term in A.P is given by[tex]a_n=a_1+(n-1)d[/tex]since we have to find the 105th term, so we can write    [tex]a_{105}=\dfrac{1}{2}+(105-1)\dfrac{1}{2}[/tex]               [tex]=\dfrac{1}{2}+\dfrac{104}{2}[/tex]               [tex]=\dfrac{105}{2}[/tex]Hence, the 105th term of given series of A.P is [tex]\dfrac{105}{2}[/tex].