Question

If $\frac{1^3+2^3+3^3+\ldots \text { up to } n \text { terms }}{1 \cdot 3+2 \cdot 5+3 \cdot 7+\ldots \text { up to } n \text { terms }}=\frac{9}{5}$, then the value of $n$ is

Answer 1

Correct Answer: 5

Step 1: Understanding the given summations 

We are given two summations:

1. Sum of cubes of the first \(n\) natural numbers:

\[ 1^3 + 2^3 + 3^3 + \dots + n^3 = \left(\frac{n(n + 1)}{2}\right)^2 \]

2. Sum of the series \( 1 \cdot 3 + 2 \cdot 5 + 3 \cdot 7 + \dots \) where the \(r\)-th term is \( r(2r + 1) \):

\[ \sum_{r=1}^{n} r(2r + 1) = \sum_{r=1}^{n} (2r^2 + r). \] 

Step 2: Simplifying the summation for the denominator

We split the summation into two parts:

\[ \sum_{r=1}^{n} (2r^2 + r) = 2 \sum_{r=1}^{n} r^2 + \sum_{r=1}^{n} r. \]

Using standard summation formulas:

\[ \sum_{r=1}^{n} r^2 = \frac{n(n + 1)(2n + 1)}{6}, \quad \sum_{r=1}^{n} r = \frac{n(n + 1)}{2}. \]

Substituting into the denominator:

\[ \sum_{r=1}^{n} (2r^2 + r) = 2 \cdot \frac{n(n + 1)(2n + 1)}{6} + \frac{n(n + 1)}{2}. \] Simplifying this expression: \[ \sum_{r=1}^{n} (2r^2 + r) = \frac{n(n + 1)(2n + 1)}{3} + \frac{n(n + 1)}{2}. \] Combining the terms: \[ \sum_{r=1}^{n} (2r^2 + r) = \frac{n(n + 1)}{6} \left( 4(2n + 1) + 3 \right) = \frac{n(n + 1)}{6} (8n + 7). \] 

Step 3: Setting up the equation

The given equation becomes:

\[ \frac{\left( \frac{n(n + 1)}{2} \right)^2}{\frac{n(n + 1)}{6} (8n + 7)} = \frac{9}{5}. \] Simplifying the fraction: \[ \frac{\left( \frac{n(n + 1)}{2} \right)^2}{\frac{n(n + 1)}{6} (8n + 7)} = \frac{n^2(n + 1)^2}{4n(n + 1)(8n + 7)} = \frac{3n(n + 1)}{2(8n + 7)}. \] Equating this to \( \frac{9}{5} \): \[ \frac{3n(n + 1)}{2(8n + 7)} = \frac{9}{5}. \] Cross-multiplying: \[ 5 \cdot 3n(n + 1) = 9 \cdot 2(8n + 7). \] Simplifying: \[ 15n(n + 1) = 18(8n + 7). \] Expanding both sides: \[ 15n^2 + 15n = 144n + 126. \] Rearranging the terms: \[ 15n^2 - 129n - 126 = 0. \] Dividing through by 3: \[ 5n^2 - 43n - 42 = 0. \] Factoring the quadratic equation: \[ (5n + 6)(n - 5) = 0. \] Therefore, \( n = 5 \) or \( n = -\frac{6}{5} \). Since \( n \) must be a positive integer, we have: \[ n = 5. \] 

Final Answer: \( n = 5 \).