Sure! Let's solve the problem of inserting three geometric means between 4 and 324 step by step.
1. Given Values:
- The first term ([tex]\( a_1 \)[/tex]) is 4.
- The fifth term ([tex]\( a_5 \)[/tex]) is 324.
- We need to insert three geometric means between these values. Hence, the total number of terms is 5 (including 4 and 324).
2. Determine the common ratio [tex]\( r \)[/tex]:
- In a geometric sequence, each term is multiplied by the common ratio to get the next term.
- The geometric sequence formula for [tex]\( a_n \)[/tex] is given by [tex]\( a_n = a_1 \times r^{(n-1)} \)[/tex].
- For the fifth term ([tex]\( a_5 = 324 \)[/tex]):
[tex]\[
a_5 = a_1 \times r^{4}
\][/tex]
[tex]\[
324 = 4 \times r^{4}
\][/tex]
3. Solve for the common ratio [tex]\( r \)[/tex]:
- Rearrange the equation to isolate [tex]\( r \)[/tex]:
[tex]\[
r^{4} = \frac{324}{4}
\][/tex]
[tex]\[
r^{4} = 81
\][/tex]
- Taking the fourth root of both sides:
[tex]\[
r = \sqrt[4]{81} = 3
\][/tex]
4. Calculate the successive terms:
- [tex]\( a_2 = a_1 \times r \)[/tex]:
[tex]\[
a_2 = 4 \times 3 = 12
\][/tex]
- [tex]\( a_3 = a_2 \times r \)[/tex]:
[tex]\[
a_3 = 12 \times 3 = 36
\][/tex]
- [tex]\( a_4 = a_3 \times r \)[/tex]:
[tex]\[
a_4 = 36 \times 3 = 108
\][/tex]
Thus, the three geometric means between 4 and 324 are 12, 36, and 108. Our sequence is:
[tex]\[ 4, 12, 36, 108, 324 \][/tex]
