To find the geometric means inserted between two given numbers, we use the properties of geometric sequences. In a geometric sequence, each term is a constant multiple (common ratio) of the previous term.
Here’s how we can identify the geometric means between 5 and [tex]\( x \)[/tex] given that the third mean is 40:
1. Identify the Given and Required:
- First term [tex]\( a = 5 \)[/tex]
- Third geometric mean [tex]\( = 40 \)[/tex]
- Last term [tex]\( x = 160 \)[/tex]
2. Establish the Sequence:
- The sequence will start at 5 and end at [tex]\( x \)[/tex]. Between these, we have 4 geometric means.
- Therefore, the full sequence will be: [tex]\( 5, GM_1, GM_2, GM_3, GM_4, x \)[/tex].
3. Set Up the Geometric Ratio:
- We know the third geometric mean [tex]\( GM_3 = 40 \)[/tex].
- Let the common ratio be [tex]\( r \)[/tex].
4. Express the Terms in Terms of [tex]\( r \)[/tex]:
- [tex]\( GM_1 = 5r \)[/tex]
- [tex]\( GM_2 = 5r^2 \)[/tex]
- [tex]\( GM_3 = 5r^3 \)[/tex]
- [tex]\( GM_4 = 5r^4 \)[/tex]
- Last term [tex]\( x = 5r^5 \)[/tex]
5. Use the Given Information to Find [tex]\( r \)[/tex]:
- Given [tex]\( GM_3 = 40 \)[/tex], substitute the value:
[tex]\[
5r^3 = 40 \implies r^3 = \frac{40}{5} \implies r^3 = 8 \implies r = 2
\][/tex]
6. Determine the Value of [tex]\( x \)[/tex]:
- Since [tex]\( x = 5r^5 \)[/tex]:
[tex]\[
x = 5 \cdot 2^5 = 5 \cdot 32 = 160
\][/tex]
7. Verify the Geometric Sequence and Find the Means:
- With [tex]\( r = 2 \)[/tex]:
- [tex]\( GM_1 = 5r = 5 \times 2 = 10 \)[/tex]
- [tex]\( GM_2 = 5r^2 = 5 \times 2^2 = 5 \times 4 = 20 \)[/tex]
- [tex]\( GM_3 = 5r^3 = 5 \times 2^3 = 5 \times 8 = 40 \)[/tex]
- [tex]\( GM_4 = 5r^4 = 5 \times 2^4 = 5 \times 16 = 80 \)[/tex]
- Last term [tex]\( x \)[/tex]:
Thus, the four geometric means between 5 and 160 are [tex]\( 10, 20, 40, \)[/tex] and [tex]\( 80 \)[/tex].
Therefore, the value of [tex]\( x \)[/tex] is [tex]\( 160 \)[/tex].
