Solution:3⁹ + 3¹² + 3¹⁵ + 3ᵐ⁺⁹ = 3⁹ × (1 + 3³ + 3⁶ + 3ᵐ)(where, m = n - 9 )
Since, 3⁹ is a cube of 3³ , let we find the value of 3ᵐ which will result in a cube for expression
(1 + 3³ + 3⁶ + 3ᵐ) = (1 + 27 + 729 + 3ᵐ)
When m = 1 ⇒ 757 + 3¹ = 760
This is not a cube.
When m = 2 ⇒ 757 + 3² = 766
This is not a cube.
When m = 3 ⇒ 757 + 3³ = 784
This is not a cube.
When n = 4 ⇒ 757 + 3⁴ = 838
This is not a cube.
When n = 5 ⇒ 757 + 3⁵ = 1000
This is (10)³
Hence, m = 5 ⇒ n = 9 + m = 14