If the 2nd term of a GP is 12 and the 4th term is 48, what is the common ratio?

If the 2nd term of a GP is 12 and the 4th term is 48, 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 12, so we have the equation: ar = 12.
Step 3: Write the formula for the 4th term of the GP, which is 'ar^3'. We know from the question that this equals 48, so we have the equation: ar^3 = 48.
Step 4: Now we have two equations: ar = 12 and ar^3 = 48.
Step 5: To find the common ratio 'r', divide the second equation (ar^3 = 48) by the first equation (ar = 12). This gives us: (ar^3) / (ar) = 48 / 12.
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) – A sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
Term Identification – Understanding how to identify and express terms in a geometric progression using the first term and common ratio.
Algebraic Manipulation – The ability to manipulate equations to isolate variables and solve for unknowns.