If the 2nd term of a GP is 8 and the 4th term is 32, what is the common ratio?

If the 2nd term of a GP is 8 and the 4th term is 32, what is the common ratio?

Step 1: Identify the first term of the geometric progression (GP) as 'a' and the common ratio as 'r'.
Step 2: Write the formula for the 2nd term of the GP, which is 'ar'. We know from the question that this equals 8, so we have the equation: ar = 8.
Step 3: Write the formula for the 4th term of the GP, which is 'ar^3'. We know from the question that this equals 32, so we have the equation: ar^3 = 32.
Step 4: Now we have two equations: ar = 8 and ar^3 = 32.
Step 5: To find the common ratio 'r', divide the second equation (ar^3 = 32) by the first equation (ar = 8). This gives us: (ar^3) / (ar) = 32 / 8.
Step 6: Simplifying the left side, we get r^2 = 4.
Step 7: To find 'r', take the square root of both sides. This gives us r = 2.

Geometric Progression (GP) – Understanding the properties of geometric sequences, including how to express terms in relation to the first term and common ratio.
Algebraic Manipulation – Skills in manipulating equations to isolate variables and solve for unknowns.