Question

The sum of first $n$ terms of a G.P. is 1023. If the first term is 1 and the common ratio is 2, then the value of $n$ is

• A. 12
• B. 11
• C. 10
• D. 9
• E. 8

Hint:

Powers of 2 like $2^{10}=1024$ are frequently used in competitive math; memorizing them saves time.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We are given the sum of the first \( n \) terms of a Geometric Progression (G.P.), the first term (\( a \)), and the common ratio (\( r \)). We need to find the number of terms, \( n \).
Step 2: Key Formula or Approach:
The formula for the sum of the first \( n \) terms of a G.P. is given by:
\[ S_n = \frac{a(r^n - 1)}{r - 1} \] where \( S_n \) is the sum, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms. This formula is used when \( r \neq 1 \).
Step 3: Detailed Explanation:
We are given the following values:
Sum, \( S_n = 1023 \)
First term, \( a = 1 \)
Common ratio, \( r = 2 \)
Substitute these values into the sum formula:
\[ 1023 = \frac{1(2^n - 1)}{2 - 1} \] Simplify the denominator:
\[ 1023 = \frac{2^n - 1}{1} \] \[ 1023 = 2^n - 1 \] Now, we need to solve for \( n \). Add 1 to both sides of the equation:
\[ 1023 + 1 = 2^n \] \[ 1024 = 2^n \] To find \( n \), we need to express 1024 as a power of 2. It's a common power of 2 to remember for competitive exams.
We know that \( 2^{10} = 1024 \).
Therefore, by comparing the exponents:
\[ n = 10 \] Step 4: Final Answer:
The value of n is 10.