If the sum of the first n terms of a GP is 63 and the first term is 7 with a com

If the sum of the first n terms of a GP is 63 and the first term is 7 with a common ratio of 2, what is n?

If the sum of the first n terms of a GP is 63 and the first term is 7 with a common ratio of 2, what is n?

Step 1: Identify the given values. The first term (a) is 7, the common ratio (r) is 2, and the sum of the first n terms (S_n) is 63.
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 = 7(1 - 2^n) / (1 - 2).
Step 4: Simplify the denominator: 1 - 2 = -1, so the equation becomes 63 = 7(1 - 2^n) / -1.
Step 5: Multiply both sides by -1 to eliminate the negative sign: -63 = 7(1 - 2^n).
Step 6: Divide both sides by 7: -63 / 7 = 1 - 2^n, which simplifies to -9 = 1 - 2^n.
Step 7: Rearrange the equation to isolate 2^n: 2^n = 1 + 9, which simplifies to 2^n = 10.
Step 8: To find n, we need to express 10 as a power of 2. Since 2^3 = 8 and 2^4 = 16, we can estimate that n is between 3 and 4.
Step 9: Use logarithms or trial and error to find that n = 5 is the correct integer solution since 2^5 = 32, which is the closest power of 2 that fits the equation.

Geometric Progression (GP) – Understanding the formula for the sum of the first n terms of a geometric series, which involves the first term and the common ratio.
Algebraic Manipulation – Ability to rearrange and solve equations derived from the GP sum formula.