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 10th term of 40,10, 5/2, 5/8, ....
Accepted Solution
A:
Answer:[tex]a_{10} = \frac{10}{65536}[/tex]Step-by-step explanation:The first step to solving this problem is verifying if this sequence is an arithmetic sequence or a geometric sequence.This sequence is arithmetic if:[tex]a_{3} - a_{2} = a_{2} - a_{1}[/tex]We have that:[tex]a_{3} = 40, a_{2} = 10, a_{3} = \frac{5}{2}[/tex][tex]a_{3} - a_{2} = a_{2} - a_{1}[/tex][tex]\frac{5}{2} - 10 = 10 - 40[/tex][tex]\frac{-15}{2} \neq -30[/tex]This is not an arithmetic sequence.This sequence is geometric if:[tex]\frac{a_{3}}{a_{2}} = \frac{a_{2}}{a_{1}}[/tex][tex]\frac{\frac{5}[2}}{10} = \frac{10}{40}[/tex][tex]\frac{5}{20} = \frac{1}{4}[/tex][tex]\frac{1}{4} = \frac{1}{4}[/tex]This is a geometric sequence, in which:The first term is 40, so [tex]a_{1} = 40[/tex]The common ratio is [tex]\frac{1}{4}[/tex], so [tex]r = \frac{1}{4}[/tex].We have that:[tex]a_{n} = a_{1}*r^{n-1}[/tex]The 10th term is [tex]a_{10}[/tex]. So:[tex]a_{10} = a_{1}*r^{9}[/tex][tex]a_{10} = 40*(\frac{1}{4})^{9}[/tex][tex]a_{10} = \frac{40}{262144}[/tex]Simplifying by 4, we have:[tex]a_{10} = \frac{10}{65536}[/tex]