Question

Observe the following pattern:
1³ = 1
1³ + 2³ = (1 + 2)²
1³ + 2³ + 3³ = (1 + 2 + 3)²
Write the next three rows and calculate the value of 1³ + 2³ + 3³ + ... + 9³ + 10³ by the above pattern.

Answer 1

Extend the pattern as follows:

$$\begin{gathered} 1^3=1 \\ {1}^3+2^3={\left(1+2\right)}^2 \\ {1}^3+2^3+3^3={\left(1+2+3\right)}^2 \\ {1}^3+2^3+3^3+4^3={\left(1+2+3+4\right)}^2 \\ {1}^3+2^3+3^3+4^3+5^3={\left(1+2+3+4+5\right)}^2 \\ {1}^3+2^3+3^3+4^3+5^3+6^3={\left(1+2+3+4+5+6\right)}^2 \end{gathered}$$

Now, from the above pattern, the required value is given by:

$1^3+2^3+3^3+4^3+5^3+6^3+7^3+8^3+9^3+{10}^3={\left(1+2+3+4+5+6+7+8+9+10\right)}^2={55}^2=3025$

Thus, the required value is 3025.