Question

Let \( \alpha, \beta (\alpha \neq \beta) \) be the values of m, for which the equations \(x + y + z = 1\), \(x + 2y + 4z = m\), and \(x + 4y + 10z = m^2\) have infinitely many solutions. Then the value of \(\sum_{n=1}^{10} (n^4 + n^8)\) is equal to:

• A. 440
• B. 3080
• C. 3410
• D. 560

Hint:

For summation problems involving polynomial powers, utilize known summation formulas efficiently.

Answer 1

The Correct Option is A

For infinite solutions based on determinant conditions: \[\Delta = \begin{vmatrix} 1 & 1 & 1
1 & 2 & 4
1 & 4 & 10 \end{vmatrix} = 4 - 6 + 2 = 0\] The valid values for m are 1 and 2. Applying the provided summation: \[\sum_{n=1}^{10} (n^4 + n^8) = \sum_{n=1}^{10} n^4 + \sum_{n=1}^{10} n^8 = 55 + 385 = 440\]