Summing Consecutive Powers

Algebra Level 3

$\large 3^2 + 4^2 = 5^2$

The above shows that there exist 3 consecutive positive integers such that the sum of squares of two of them is equal to the square of the last number.

Likewise, does there exist 4 consecutive positive integers such that the sum of cubes of three of them is equal to the cube of the last number?

No Yes

1 solution

Maria Kozlowska
Dec 3, 2016

$a^3 + (a+1)^3 + (a+2)^3 = (a+3)^3 \Rightarrow a³ - 6a - 9 = 0 \Rightarrow a(a^2-6)=3 \times 3 \Rightarrow a=3$

$3^3 + 4^3 + 5^3=6^3$

Bonus: Find all positive integers $n\geq 1$ such that there exist $n+1$ consecutive positive integers for which the sum of $n^\text{th}$ perfect power of $n$ of them is equal to the $n^\text{th}$ perfect power of the last number?

Pi Han Goh - 4 years, 6 months ago

Summing Consecutive Powers

1 solution

0 pending reports