Question

1 cube + 2 cube + 3 cube + .....+ k cube = 16900, then find 1+2+3+......+k

Answer 1

The value of 1 + 2 + 3 +...... + k = 130

Step-by-step explanation:

We have,

$1^{3} +2^{3} +3^{3} +..................+ k^3$ = 16900          

To find, the value of 1 + 2 + 3 +...... + k = ?

We know that,

$1^{3} +2^{3} +3^{3} +..................+ n^3$ = $(\dfrac{n(n+1)}{2})^2$

and

1 + 2 + 3 +...... + n = $\dfrac{n(n+1)}{2}$

∴ 1 + 2 + 3 +...... + k = $\dfrac{k(k+1)}{2}$       ........... (1)

Also, $1^{3} +2^{3} +3^{3} +..................+ k^3$ = 16900      

⇒ $(\dfrac{k(k+1)}{2})^2$ = 16900  

⇒ $(\dfrac{k(k+1)}{2})^2$ = $130^2$

⇒ $\dfrac{k(k+1)}{2}$ = 130

Now, equation (1) can be written as

1 + 2 + 3 +...... + k = 130

∴ The value of 1 + 2 + 3 +...... + k = 130