The first term of a geometric sequence is 16 and the fifth term is 9. What is the value of the seventh term?

To find the seventh term of a geometric sequence given the first term and the fifth term, you can follow these steps:

Step 1: Identify the first term and the fifth term

The first term, (a_1), is 16, and the fifth term, (a_5), is 9.

Step 2: Use the geometric sequence formula to find the common ratio (r)

The formula for the nth term of a geometric sequence is: [ a_n = a_1 \cdot r^{n-1} ]

For the fifth term: [ a_5 = a_1 \cdot r^{5-1} ] [ 9 = 16 \cdot r^4 ]

Step 3: Solve for (r)

[ 9 = 16 \cdot r^4 ] [ \frac{9}{16} = r^4 ] [ r^4 = \frac{9}{16} ] Taking the fourth root of both sides: [ r = \left(\frac{9}{16}\right)^{\frac{1}{4}} ] [ r = \left(\frac{3^2}{4^2}\right)^{\frac{1}{4}} ] [ r = \left(\frac{3}{4}\right)^{\frac{1}{2}} ] [ r = \frac{3}{4} \text{ or } \frac{-3}{4} ] However, since the sequence goes from 16 to 9, the ratio must be positive, so: [ r = \frac{3}{4} ]

Step 4: Use the common ratio to find the seventh term

Now, use the formula for the nth term with (n = 7): [ a_7 = a_1 \cdot r^{7-1} ] [ a_7 = 16 \cdot \left(\frac{3}{4}\right)^6 ]

Step 5: Calculate the seventh term

[ a_7 = 16 \cdot \left(\frac{3}{4}\right)^6 ] [ a_7 = 16 \cdot \left(\frac{3^6}{4^6}\right) ] [ a_7 = 16 \cdot \left(\frac{729}{4096}\right) ] [ a_7 = 16 \cdot \frac{729}{4096} ] [ a_7 = \frac{11664}{4096} ] [ a_7 = \frac{729}{256} ] [ a_7 = 2.85 \text{ (approximately, but for exactness)} ] [ a_7 = \frac{729}{256} ]

So, the exact value of the seventh term is (\frac{729}{256})45.