Final answer:
To find the common ratio of a geometric sequence with a first term of 8 and a sum of the first three terms being 78, set up and solve the quadratic equation derived from the sum formula for the first three terms of the sequence.
Step-by-step explanation:
The student is asking to find the common ratio of a geometric sequence where the first term is 8 and the sum of the first three terms is 78. To solve for the common ratio (r), we can set up equations based on the definition of a geometric sequence: a_1 = 8, a_2 = 8r, a_3 = 8r^2, and the sum of these terms is 78.
The sum of the first three terms can be expressed as: S_3 = a_1 + a_2 + a_3 = 8 + 8r + 8r^2 = 78. Simplifying gives: 8(1 + r + r^2) = 78. To find the common ratio, divide both sides by 8: 1 + r + r^2 = 9.75.
Now the student can use technology, like a graphing calculator or algebraic software, to solve the quadratic equation r^2 + r - 8.75 = 0. They can either graph the equation and find the x-intercepts or use the quadratic formula to find the values of r. Since a geometric sequence has a common ratio that is a constant multiplier, the student looks for the positive value when solving this quadratic equation.