Find the fifth term in the geometric sequence, given the first term and the common ratio.
[tex]
\begin{array}{c}
a_1=243 \text { and } r=\frac{1}{3} \
a_5=[?]
\end{array}
[/tex]
Answer (2)
The formula for the nth term of a geometric sequence is a n = a 1 n − 1 .
Substitute the given values: a 1 = 243 , r = f r a c 1 3 , and n = 5 .
Calculate $a_5 = 243 ( frac{1}{3})^{4} = 243
frac{1}{81}$.
Simplify to find the fifth term: a 5 = 3 .
Explanation
Understanding the Problem We are given a geometric sequence with the first term a 1 = 243 and a common ratio r = 3 1 . We want to find the fifth term, a 5 .
Recall the Formula The formula for the n th term of a geometric sequence is given by: a n = a 1 ⋅ r n − 1 where a 1 is the first term, r is the common ratio, and n is the term number.
Substitute the Values In our case, we want to find the fifth term, so n = 5 . We are given a 1 = 243 and r = 3 1 . Plugging these values into the formula, we get: a 5 = 243 ⋅ ( 3 1 ) 5 − 1 = 243 ⋅ ( 3 1 ) 4
Calculate the Power Now, we calculate ( 3 1 ) 4 :
( 3 1 ) 4 = 3 4 1 4 = 81 1
Simplify the Expression Next, we multiply 243 by 81 1 :
a 5 = 243 ⋅ 81 1 = 81 243 Since 243 = 3 ⋅ 81 , we have: a 5 = 81 3 ⋅ 81 = 3
State the Answer Therefore, the fifth term of the geometric sequence is 3.
Examples
Geometric sequences are useful in many real-world applications, such as calculating compound interest, modeling population growth or decay, and determining the trajectory of a bouncing ball. For example, if a ball is dropped from a height of 243 feet and bounces to one-third of its previous height with each bounce, the height of the ball after the fifth bounce would be the fifth term of the geometric sequence we just calculated. This helps engineers design equipment and predict outcomes in various scenarios.
The fifth term in the geometric sequence with first term 243 and common ratio 3 1 is calculated to be 3 using the formula for the nth term of a geometric sequence. By substituting the values and simplifying, we find that a 5 = 3 .
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