Final answer:
To find the first three terms of a geometric sequence with a first term of 9 and a given ratio, we can use the formula for the sum of a geometric sequence. Solving the equation, we find that the common ratio is 3. Then, we can calculate the second term by multiplying the first term by the common ratio and the third term by multiplying the second term by the common ratio again. The first three terms of the sequence are 9, 27, and 81.
Step-by-step explanation:
To solve this problem, we can use the formula for the sum of a geometric sequence.
The sum of the first n terms of a geometric sequence with first term a and common ratio r is given by:
Sn = a * (1 - rn) / (1 - r)
Using the given information, we can set up two equations:
a * (1 - r8) / (1 - r) / (a * (1 - r4) / (1 - r)) = 97 / 81
a * (1 - r8) / (1 - r4) = 97 / 81
We also know that the first term, a, is 9. Substituting this into the equation, we get:
9 * (1 - r8) / (1 - r4) = 97 / 81
Now, we can solve for r by cross multiplying and simplifying the equation:
729 - 729r8 = 8718 - 8718r4
729r8 - 8718r4 - 7989 = 0
By factoring the equation, we find that (r - 3)(729r7 + 2187r6 + 6561r5 + 19683r4 + 59049r3 + 177147r2 + 531441r + 1594323) = 0.
Since all terms are positive, we can ignore the second factor.
So, r = 3. Now, we can find the second term by multiplying the first term by the common ratio:
a2 = a * r = 9 * 3 = 27
Finally, the third term can be found by multiplying the second term by the common ratio again:
a3 = a2 * r = 27 * 3 = 81