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:

\[1^3 = 1\]

\[ 1^3 + 2^3 = \left( 1 + 2 \right)^2 \]

\[ 1^3 + 2^3 + 3^3 = \left( 1 + 2 + 3 \right)^2 \]

\[ 1^3 + 2^3 + 3^3 + 4^3 = \left( 1 + 2 + 3 + 4 \right)^2 \]

\[ 1^3 + 2^3 + 3^3 + 4^3 + 5^3 = \left( 1 + 2 + 3 + 4 + 5 \right)^2 \]

\[ 1^3 + 2^3 + 3^3 + 4^3 + 5^3 + 6^3 = \left( 1 + 2 + 3 + 4 + 5 + 6 \right)^2 \]

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.