Question

\(\frac{\text{2+3+5+…. to n terms}}{\text{2+5+8+….8 \,to \,8 \,\,terns}}\)=9, then the value of n is

• A. 20
• B. 30
• C. 10
• D. 40

Answer 1

The Correct Option is D

We are tasked with solving the equation:

\[ \frac{\text{2 + 3 + 5 + … to } n \text{ terms}}{\text{2 + 5 + 8 + … to 8 terms}} = 9, \]

and finding the value of \( n \).

Step 1: Analyze the numerator and denominator.

The numerator is the sum of the first \( n \) terms of an arithmetic progression (AP) with first term \( a_1 = 2 \) and common difference \( d_1 = 1 \). The denominator is the sum of the first 8 terms of another AP with first term \( a_2 = 2 \) and common difference \( d_2 = 3 \).

Step 2: Use the formula for the sum of an AP.

The sum of the first \( k \) terms of an AP is given by:

\[ S_k = \frac{k}{2} \left( 2a + (k - 1)d \right). \]

Step 3: Compute the denominator.

For the denominator, \( k = 8 \), \( a_2 = 2 \), and \( d_2 = 3 \):

\[ S_8 = \frac{8}{2} \left( 2(2) + (8 - 1)(3) \right) = 4 \left( 4 + 21 \right) = 4 \times 25 = 100. \]

Step 4: Express the numerator.

For the numerator, \( k = n \), \( a_1 = 2 \), and \( d_1 = 1 \):

\[ S_n = \frac{n}{2} \left( 2(2) + (n - 1)(1) \right) = \frac{n}{2} \left( 4 + n - 1 \right) = \frac{n}{2} \left( n + 3 \right). \]

Step 5: Set up the equation.

Substitute the expressions for the numerator and denominator into the given equation:

\[ \frac{\frac{n}{2} \left( n + 3 \right)}{100} = 9. \]

Simplify:

\[ \frac{n(n + 3)}{200} = 9. \]

Multiply through by 200:

\[ n(n + 3) = 1800. \]

Step 6: Solve the quadratic equation.

Rearrange into standard quadratic form:

\[ n^2 + 3n - 1800 = 0. \]

Factorize:

\[ (n + 45)(n - 40) = 0. \]

Thus, \( n = -45 \) or \( n = 40 \). Since \( n \) must be positive, we have:

\[ n = 40. \]

Final Answer: The value of \( n \) is \( \mathbf{40} \).