If the sum of the first n terms of a GP is 63 and the first term is 3, what is t

If the sum of the first n terms of a GP is 63 and the first term is 3, what is the common ratio if n = 4?

If the sum of the first n terms of a GP is 63 and the first term is 3, what is the common ratio if n = 4?

Step 1: Identify the given values. The first term (a) is 3, the sum of the first n terms (S_n) is 63, and n is 4.
Step 2: Write down the formula for the sum of the first n terms of a geometric progression (GP): S_n = a(1 - r^n) / (1 - r).
Step 3: Substitute the known values into the formula: 63 = 3(1 - r^4) / (1 - r).
Step 4: Multiply both sides by (1 - r) to eliminate the fraction: 63(1 - r) = 3(1 - r^4).
Step 5: Distribute on both sides: 63 - 63r = 3 - 3r^4.
Step 6: Rearrange the equation to bring all terms to one side: 3r^4 - 63r + 60 = 0.
Step 7: Simplify the equation by dividing everything by 3: r^4 - 21r + 20 = 0.
Step 8: Factor the equation or use the quadratic formula to find the value of r. In this case, we can find that r = 2.

Geometric Progression (GP) – Understanding the properties and formulas related to geometric progressions, including the sum of the first n terms.
Sum of a GP – Application of the formula for the sum of the first n terms of a geometric series.
Algebraic Manipulation – Skills in rearranging and solving equations to find unknown variables.